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the idea of studying `` network motifs '' in biological systems , namely simple , frequently occurring , structures from which some properties of the system may be inferred , has attracted considerable recent interest ( for example @xcite ) . the reason is quite natural : systems biology proceeds dually by elucidating the functioning of subsystems while at the same time attempting to fit together these pieces to explain large - scale behaviours in biological systems . at the heart of this study is a need to know how behaviours in a dynamical system are affected by embedding this system in some larger system in well - defined ways . some rigorous results in this area come from the literature on hyperbolicity and structural stability ( @xcite , for example ) , and on monotone control systems ( @xcite , for example ) . a closely related type of question is when a behaviour known to occur in some member of a _ family _ of dynamical systems @xmath0 can be guaranteed to occur in some member of a new family @xmath1 related to @xmath0 in some natural sense . our goal here is understanding the inheritance of _ multiple positive nondegenerate equilibria _ ( mpne , definition  [ defmpne ] below ) and _ multiple positive linearly stable equilibria _ ( mpse , definition  [ defmpse ] below ) in chemical reaction networks ( crns ) which form the main component of most biological models . the literature on multiple equilibria in crns is extensive and we do not attempt to provide a bibliography ; the recent papers @xcite , @xcite , and @xcite provide some references to key strands of this work . but most work , including our own , has focussed on conditions for _ precluding _ multistationarity ; in some ways the opposite of the question treated here . an exception is the paper by joshi and shiu @xcite , which inspired this paper . the authors treated questions of inheritance of multistationarity primarily , but not exclusively , focussed on _ fully open _ crns ( definition  [ deffullyopen ] below ) . they highlighted that in studying multistationarity ( and potentially other nontrivial behaviours in crns ) , we need to identify a relevant partial order on the set of all crns , and then search for minimal elements w.r.t . this partial order , which they termed _ atoms_. this partial order @xmath2 should be such that if some crn @xmath3 displays the behaviour of interest , and @xmath4 , then @xmath5 must display this behaviour . in the context of multistationarity in fully open crns , and with certain kinetic assumptions , there is a nice result : if we identify crns with their natural digraphs ( see definition  [ defpetrinet ] below ) , then mpne and mpse are inherited in the induced subnetwork partial order ( see definition  [ defiso ] ) . this result ( corollary  [ coropeninduced ] below ) was largely known to joshi and shiu ( corollary  4.6 of @xcite ) . in fact , the results here allow considerably stronger claims about the inheritance of mpne and mpse in fully open networks ( see remark  [ remfobest ] ) .    results about fully open crns do not generalise easily to arbitrary crns : a crn which forbids multistationarity may have induced subnetworks which permit multistationarity . building larger crns from smaller ones in natural ways such as adding reactions can quite generally both introduce and destroy multistationarity . nevertheless , it is possible to formulate several operations which together define a partial order @xmath6 on crns such that if @xmath3 displays mpne ( resp . , mpse ) and @xmath7 then @xmath5 displays mpne ( resp . , mpse ) , and it is this task which we _ begin _ in this paper . for example , we can add new reactions involving new species while preserving mpne and mpse provided certain conditions are met ( theorem  [ thmblockadd ] ) . feliu and wiuf @xcite show that under various circumstances multistationarity survives when a reaction is `` split '' with the introduction of a new intermediate species . we have a result involving splitting reactions and inserting intermediate complexes ( theorem  [ thmintermediates ] ) which reproduces and generalises some elements of their results . availability of a partial order relevant to mpne or mpse couples naturally with the task of characterising minimal networks admitting these behaviours w.r.t . this partial order . although minimal networks need not be small , in the sense of having few species , few reactions , or limited numbers of molecules participating in each reaction , a first step is naturally to identify _ small _ minimal subnetworks with the desired behaviour . we do not undertake this task here , but it was begun for the induced subnetwork partial order in @xcite and continued in the recent preprint @xcite .    our proofs all follow roughly the same template , relying heavily on the implicit function theorem , with some insight from differential topology . we do not need to appeal to degree theory @xcite , or homotopy theory @xcite , or any nontrivial algebra , even though many of the resulting claims are algebraic in the sense of being about the zeros of polynomial equations . our local approach , though powerful , necessarily has limitations discussed in remark  [ remlimit ] and the conclusions . the paper is laid out as follows . after introducing some basic notions in section  [ secbackground ] we present an extended example ( section  [ secextended ] ) beginning with a crn which displays mpne and illustrating some of the many ways one might `` build up '' a reaction network , some of which will be proved to preserve mpne and some of which destroy it . this foreshadows the theorems to follow , while also highlighting their limitations . we then present in section  [ secthms ] the results , some known and some new , but defer the proofs until section  [ secproofs ] , after illustrating application of the results on a biologically important crn in section  [ secbioexample ] . a consequence of the analysis in section  [ secbioexample ] is previewed in figure  [ fig : mapk ] . finally , we present some discussion and conclusions . we provide only a brief , informal introduction to crns , and the reader is referred to @xcite for a more thorough discussion in a closely related framework . @xmath8 refers to the nonnegative orthant in @xmath9 , namely @xmath10 , while @xmath11 refers to the positive orthant , namely @xmath12 . for @xmath13 , @xmath14 will mean that @xmath15 . a vector is nonnegative ( resp . , positive ) if each component is nonnegative ( resp . , positive ) . consider a set of @xmath16 chemical species @xmath17 with concentrations @xmath18 , and a crn involving @xmath19 reactions between these species . let the system have _ stoichiometric matrix _ @xmath20 with rank @xmath21 . here @xmath20 is an @xmath22 real matrix , and @xmath23 is the net change in concentration of species @xmath24 when reaction @xmath25 occurs . the @xmath25th column of @xmath20 is the _ reaction vector _ for the @xmath25th reaction . in spatially homogeneous , deterministic , continuous time models , the evolution of the species concentrations is governed the ode : @xmath26 where @xmath27 is the vector of reaction rates . we assume that @xmath27 is defined and @xmath28 on @xmath11 . it is natural to put some further minimal assumptions on @xmath27 , so we always assume that @xmath27 belongs to the class of _ positive general kinetics _ defined formally in @xcite : in brief , the rate of an irreversible reaction ( i ) is positive on the positive orthant ; ( ii ) depends only on the concentrations of its substrates ; and ( iii ) increases strictly with concentration of each substrate on the positive orthant . restrictions on the functions describing the rate of a reversible reaction are easily derived from the rules for irreversible reactions . positive general kinetics includes mass action ( ma ) kinetics as a special case , and in fact all our inheritance results are consistent with ma kinetics . in other words , we choose ma kinetics for any added reactions , and where a reaction is modified in some way , if the original reaction had ma kinetics , then so does the new one . throughout the text , we include remarks about other classes of kinetics for which the results hold  often requiring only minor , formal modification of the proofs . a complex is a formal linear combination of species . if the species of a crn are @xmath17 , and @xmath29 is a nonnegative integer vector in @xmath9 , we write @xmath30 for the complex @xmath31 . the zero complex @xmath32 will be denoted @xmath33 . [ defpetrinet ] consider a crn @xmath3 , each of whose reactions is considered to be irreversible ( namely reversible reactions are considered as pairs of reactions ) . the _ petri net graph ( pn graph ) _ of @xmath3 , denoted @xmath34 , and also known as the species - reaction petri net of @xmath3 @xcite , is an edge - weighted bipartite digraph . its two vertex sets @xmath35 ( species vertices ) and @xmath36 ( reaction vertices ) correspond to the species and the reactions of @xmath3 , and given @xmath37 and @xmath38 , there exists an arc @xmath39 ( resp . , @xmath40 ) with weight @xmath41 if and only if the species corresponding to @xmath42 occurs with stoichiometry @xmath41 as a substrate ( resp . , product ) in reaction corresponding to @xmath27 . arcs are labelled with the stoichiometry with which the species figures in the reaction ( with the convention that labels of @xmath43 are omitted ) . petri net graphs provide one natural way of visualising and analysing crns . they are closely related to various other bipartite graphs associated with crns , particularly sr and dsr graphs @xcite . we illustrate below the pn graph of @xmath44 , @xmath45 :    ( 1,0.5 ) circle ( 1.5pt ) ; ( 1,-0.5 ) circle ( 1.5pt ) ; ( 3,0.5 ) circle ( 1.5pt ) ; ( 3,-0.5 ) circle ( 1.5pt ) ;    at ( 0,0 ) @xmath33 ; at ( 2,0 ) @xmath46 ; at ( 4,0 ) @xmath47 ;    ( 0.1,0.1 ) .. controls ( 0.3,0.3 ) and ( 0.5,0.5 ) .. ( 0.9,0.5 ) ; ( 1.85,0.15 ) .. controls ( 1.7,0.3 ) and ( 1.5,0.5 ) .. ( 1.1,0.5 ) ; ( 1.85,-0.15 ) .. controls ( 1.7,-0.3 ) and ( 1.5,-0.5 ) .. ( 1.1,-0.5 ) ; ( 0.1,-0.1 ) .. controls ( 0.3,-0.3 ) and ( 0.5,-0.5 ) .. ( 0.9,-0.5 ) ; ( 2.15,0.15 ) .. controls ( 2.3,0.3 ) and ( 2.5,0.5 ) .. ( 2.9,0.5 ) ; ( 3.85,0.15 ) .. controls ( 3.7,0.3 ) and ( 3.5,0.5 ) .. ( 3.1,0.5 ) ; ( 2.15,-0.15 ) .. controls ( 2.3,-0.3 ) and ( 2.5,-0.5 ) .. ( 2.9,-0.5 ) ; ( 3.85,-0.15 ) .. controls ( 3.7,-0.3 ) and ( 3.5,-0.5 ) .. ( 3.1,-0.5 ) ; ( 3.85,0.05 ) .. controls ( 3.5 , 0.05 ) and ( 3.3 , 0.2 ) .. ( 3.07,0.43 ) ; at ( 3.35,0.05 ) @xmath48 ; at ( 3.6,0.5 ) @xmath49 ;    [ defiso ] crns @xmath50 and @xmath51 are _ isomorphic _ if @xmath52 and @xmath53 are isomorphic as edge - labelled digraphs , via an isomorphism which preserves the given bipartition into species and reaction vertices . this is equivalent to ( independently ) relabelling species and reactions . an _ induced subnetwork _ of a crn @xmath3 is any crn whose pn graph is isomorphic to a vertex - induced subgraph of @xmath34 , namely can be obtained by deleting some subset of vertices and all their incident arcs from @xmath34 . if we say that `` @xmath50 is an induced subnetwork of @xmath51 '' , we mean that @xmath50 is isomorphic to an induced subnetwork of @xmath51 . it is straightforward that the relationship `` is an induced subnetwork '' corresponds to a partial order on the set of crns . although it is not a mathematical requirement , it is quite natural to remove reactions with no species ( isolated reaction vertices ) , species which participate in no reactions ( isolated species vertices ) , and copies of reactions , from an induced subnetwork . if we were to add these operations into the definition of an induced subnetwork we obtain precisely the definition of `` embedded networks '' in definition  2.2 of @xcite . [ deffullyopen ] a crn involving species @xmath54 is _ fully open _ if it includes all the inflow - outflow reactions @xmath55 ( @xmath56 ) . the _ fully open extension _ of a crn @xmath3 is created by adjoining to @xmath3 any such reactions which are absent from @xmath3 . @xmath57 is referred to as the _ stoichiometric subspace _ of the crn in ( [ gencrn ] ) . the nonempty intersection of a coset of @xmath57 with the nonnegative ( resp . , positive ) orthant is a _ stoichiometry class _ ( resp . , _ positive stoichiometry class _ ) of the crn . stoichiometry classes are easily shown to be invariant for ( [ gencrn ] ) under very general assumptions . we say that @xmath58 are _ compatible _ if @xmath59 . the crn evolving according to ( [ gencrn ] ) has _ multiple positive equilibria _ ( mpe ) if there exist @xmath60 s.t . @xmath59 and @xmath61 . note that it is always implicit in our use of the term `` multiple equilibria '' that the equilibria are compatible . * nondegeneracy and stability . * we say that a point @xmath62 is _ nondegenerate _ for ( [ gencrn ] ) if the jacobian determinant of the restricted vector field @xmath63 is nonzero at @xmath62 ( see also definition  4 in @xcite ) . nondegeneracy is most naturally expressed via reduced jacobian matrices and the reduced determinant of @xmath64 , defined as follows . [ notationreddet ] consider a differentiable function @xmath65 , and suppose that @xmath66 , where @xmath67 is some @xmath68-dimensonal linear subspace of @xmath9 . let @xmath69 be some matrix whose columns are a basis for @xmath67 , so there is a unique differentiable function @xmath70 s.t . ( @xmath72 is the image of @xmath73 under @xmath74 in the local coordinates on @xmath67 given by the columns of @xmath69 . ) we term @xmath75 the _ reduced jacobian matrix _ of @xmath74 w.r.t . a different choice of basis for @xmath67 , written as the columns of a matrix @xmath76 , leads to a reduced jacobian matrix @xmath77 similar to @xmath78 . when interested only in the spectral properties of such reduced jacobian matrices , we refer to the linear operator they represent as @xmath79 . we refer to @xmath80 as the _ reduced determinant _ of @xmath74 w.r.t . @xmath67 . [ defreddet ] a point @xmath62 is _ nondegenerate _ for ( [ gencrn ] ) if the reduced determinant of @xmath81 w.r.t . @xmath57 is nonzero at @xmath62 , namely @xmath82 . @xmath83 can be computed in various ways whose equivalence is shown in @xcite . most simply , @xmath83 is the sum of @xmath84 principal minors of @xmath85 where @xmath86 , namely upto a sign change , the leading nonzero coefficient in the characteristic polynomial of @xmath85 . alternatively , following definition  [ notationreddet ] , we may choose @xmath87 to be any matrix whose columns are a basis for @xmath57 and define @xmath88 via @xmath89 . then @xmath90 . [ defmpne ] a crn of the form ( [ gencrn ] ) with kinetics in some fixed class ( such as positive general kinetics or mass action kinetics ) admits _ multiple , positive , nondegenerate equilibria _ ( mpne ) if there exists _ some _ choice of allowed kinetics s.t . there exist a pair of nondegenerate equilibria on _ some _ positive stoichiometry class , namely there exist @xmath91 satisfying @xmath59 with @xmath92 and @xmath93 . we say that a square matrix @xmath94 is _ hurwitz stable _ if its eigenvalues all lie in @xmath95 , the ( open ) left half plane of @xmath96 . [ defmpse ] a crn of the form ( [ gencrn ] ) with kinetics in some fixed class admits _ multiple positive linearly stable equilibria _ ( mpse ) if there exists _ some _ choice of allowed kinetics s.t . it has two positive , compatible equilibria , both linearly stable w.r.t . their stoichiometry class , namely @xmath91 satisfying @xmath59 with @xmath97 and @xmath98 both hurwitz stable . a crn which admits mpse admits mpne by definition . * inheritance of mpne / mpse . * given a crn @xmath3 displaying mpne ( resp . , mpse ) , we will present a number of ways in which we can modify @xmath3 to create a new crn @xmath5 in such a way that @xmath5 displays mpne ( resp . , this will generally involve adding more reactions and/or more species , or modifying reactions in some way . a similar project could be undertaken for other dynamical behaviours , in particular for crns which admit hyperbolic periodic orbits , and we remark on this after certain proofs .    at the heart of our proofs is the implicit function theorem ( ift ) which , for our purposes here , takes the form :    [ lemift ] let @xmath99 be open and @xmath100 be @xmath28 . suppose @xmath101 for some @xmath102 and the jacobian matrix @xmath103 ( namely with respect to the first variables ) is nonsingular . then there exist @xmath104 , @xmath105 both open with @xmath106 , and a @xmath28 function @xmath107 satisfying @xmath108 , and @xmath109 namely the set of zeros of @xmath74 in @xmath110 is precisely the graph of @xmath111 . see , for example , theorem 9.28 in @xcite . application of the ift will frequently take the following form :    [ corift]let @xmath112 be open and let @xmath113 be a @xmath68-dimensional linear subspace of @xmath9 . let @xmath114 and consider a @xmath28 function @xmath115 s.t . for each fixed @xmath116 , @xmath117 satisfies @xmath118 . let @xmath119 be a nondegenerate zero of @xmath120 w.r.t . @xmath67 , namely , @xmath121 and @xmath122 . then there exists @xmath123 and a @xmath28 curve @xmath124 with @xmath125 and @xmath126 for @xmath127 . moreover , @xmath128 is an isolated zero of @xmath129 on @xmath130 . as in definition  [ notationreddet ] , let @xmath69 be a matrix whose columns are a basis for @xmath67 , and define @xmath131 via @xmath132 . let @xmath133 . note that @xmath47 is open and includes the origin . define @xmath134 by @xmath135 . note that ( fixing @xmath116 ) @xmath136 ( where @xmath137 ) , namely the reduced jacobian matrix of @xmath129 w.r.t . @xmath69 is just the jacobian matrix of @xmath138 w.r.t . its first argument . as @xmath62 is a nondegenerate zero of @xmath120 , we have @xmath139 and @xmath140 is nonsingular . we may then apply the ift to @xmath138 giving an open neighbourhood @xmath141 of @xmath33 in @xmath142 , some @xmath143 , and a function @xmath144 s.t . @xmath145 the result follows with @xmath146 . [ remlimit ] the results here are all * local * , and we never use global algebraic methods . a local approach , although powerful , necessarily has limitations : we will see by example in the following section that there are situations where the capacity for mpne is inherited , but we are unlikely to see this via a local argument . to motivate the results to follow and highlight their limitations we present an extended example beginning with a crn @xmath147 displaying mpne and then performing various modifications to create crns @xmath50 to @xmath148 . each modification can either be proved to preserve mpne using a result in the paper , or highlights some point about the scope of these theorems . @xmath147 appears in remark 4.4 of joshi and shiu s paper @xcite .    * base reaction network . * we begin with the following crn on two species : @xmath149 if @xmath147 is given mass action kinetics with both rate constants set at @xmath43 , we get the odes @xmath150 . positive equilibria occur when @xmath151 . the stoichiometric matrix has rank @xmath43 and the stoichiometry classes of @xmath147 satisfy @xmath152 for some constants @xmath153 . the system has reduced determinant @xmath154 ( the trace of the jacobian matrix ) . it is easy to confirm by direct calculation that any stoichiometry class with @xmath155 contains two positive equilibria at @xmath156 , @xmath157 , where @xmath158 . the reduced determinant is @xmath159 and @xmath160 at @xmath62 and @xmath161 respectively , and hence , as @xmath162 implies @xmath163 and @xmath164 , @xmath62 and @xmath161 are nondegenerate . thus @xmath147 admits mpne . that @xmath147 has mpne can also be shown as an immediate application of theorem 4.5 in @xcite . @xmath165    * modification 1 : adding new linearly dependent reactions preserves mpne . * consider adding to @xmath147 some new reaction with mass action kinetics and with reaction vector which is a linear combination of existing reaction vectors , so that the stoichiometric subspace does not change . for example we might enlarge @xmath147 with the reaction @xmath166 to get @xmath167 intuition says that provided we can make the rate of the new reaction small on any compact set , its addition represents only a small perturbation to the vector field on any compact region of some particular stoichiometry class , and hence mpne should occur for @xmath50 . this is easily formalised as a general result ( theorem  [ thmnewdepreac ] ) . the following three examples show that adding new reactions which are not linearly dependent on the old ones may have different effects on mpne .    * modification 2 : adding new independent reactions may destroy mpne . * consider now adding to @xmath147 the reversible reaction @xmath168 with mass action kinetics , to get @xmath169 the unique stoichiometry class of this system is now @xmath170 , and the unique positive equilibrium of this system is at @xmath171 . regardless of how we choose the rate constants , mpne has been destroyed . thus in general the capacity for mpne is not inherited by supernetworks of a crn .    * modification 3 : adding new independent reactions may allow multiple degenerate equilibria . * consider adding to @xmath147 the reaction @xmath172 with mass action kinetics , to get @xmath173 the unique stoichiometry class of @xmath174 is @xmath170 and the reduced determinant is easily computed to be @xmath175 . if @xmath176 , this system has no equilibria at all . on the other hand if @xmath177 , then the set of equilibria consists of solutions to @xmath178 , and all these equilibria are degenerate . thus the capacity for mpne is destroyed , even though @xmath174 still allows multiple positive _ degenerate _ equilibria for special choices of rate constants . * modification 4 : adding new independent reactions may preserve mpne . * consider adding to @xmath147 three irreversible reactions @xmath179 , @xmath180 and @xmath181 , and choosing mass action rate constants as indicated : @xmath182 the unique stoichiometry class of @xmath183 is @xmath170 . positive equilibria are solutions to @xmath184 . the hyperbola and circle have two positive intersections at @xmath185 for @xmath186 and @xmath187 , and these two points correspond to nondegenerate equilibria . thus the capacity for mpne is preserved . however , there is no obvious _ local _ argument for why mpne should be preserved in this case : the mpne constructed for the new system are not in any sense derived as perturbations of original mpne . * modification 5 : adding inflows and outflows of all species preserves mpne . * suppose we add to @xmath147 inflows and outflows of all species to get the fully open extension of @xmath147 ( definition  [ deffullyopen ] ) : @xmath188 the stoichiometric subspace becomes all of @xmath189 . this modification is well known to preserve mpne . a variety of proofs are known and a very simple ift - based argument is possible ( theorem  [ thmopenextension ] ) . @xmath165    * modification 6 : adding a trivial species preserves mpne . * suppose we add a new species @xmath190 into the reactions of @xmath147 in a rather trivial way : the species always figures with the same stoichiometry on both sides of any reaction in which it participates . we might get for example : @xmath191 the stoichiometry classes of @xmath192 are still one dimensional and are parallel to the @xmath193 plane ( namely , @xmath194 is constant ) . intuition suggests that mpne should be preserved , and this turns out to be the case . although this result ( theorem  [ thmtrivial ] ) is very easy to prove , it is quite important as a step in building crns displaying mpne from smaller ones . * modification 7 : adding new species may destroy mpne . * suppose we add a new species into the reactions of @xmath147 to get , for example , @xmath195 in this case , stoichiometry classes of @xmath196 remain one dimensional and are defined by the restrictions @xmath197 , @xmath198 , and positive equilibria occur when @xmath199 where @xmath200 is some positive constant ( a ratio of the rate constants ) . a little substitution gives that @xmath201-values of positive equilibria occur when @xmath202 , while @xmath73 and @xmath194 values are then uniquely determined . as this quadratic has exactly one positive root ( for any positive values of @xmath203 and @xmath200 ) , the capacity for mpne is destroyed . thus , in general , adding new species into reactions may destroy mpne . * modification 8 : adding new species may preserve mpne . * suppose , instead , we add the new species into existing reactions to get : @xmath204 stoichiometry classes of @xmath205 remain one dimensional and are defined by the restrictions @xmath197 , @xmath198 , and positive equilibria occur when @xmath206 where @xmath200 is some positive constant ( a ratio of the rate constants ) . choosing @xmath207 and @xmath208 , a little substitution gives that @xmath201-values of positive equilibria occur when @xmath209 . we can check , for example , that @xmath210 , and @xmath211 are positive , stoichiometrically compatible equilibria of @xmath205 . thus the capacity for mpne is preserved in this case . it remains unclear whether there is a local argument for why this should be the case . @xmath165    * modification 9 : adding new reactions involving new species often preserves mpne . * consider adding to @xmath147 a single reversible reaction @xmath212 involving a new species @xmath190 to get : @xmath213 the evolution now occurs in @xmath214 , and the stoichiometric subspace of @xmath215 is now @xmath216-dimensional . it seems plausible ( though by no means obvious ) that choosing the rate constants of the new reaction appropriately should lead to mpne in the new crn , and indeed this turns out to be the case . this is a simple case of a much more general result ( theorem  [ thmblockadd ] ) allowing the addition of several new reactions involving several new species . we remark that theorem  [ thmblockadd ] includes a nondegeneracy condition which in this case is equivalent to the requirement that the new species must not occur with the same stoichiometry on both sides of the new reaction . the reader may easily confirm , for example , that adding @xmath217 to @xmath147 , results in a crn with a unique equilibrium on each stoichiometry class for all rate constants . * modification 10 : adding a new species into reactions while also adding its inflow and outflow preserves mpne . * suppose we add into some reactions of @xmath147 a new species @xmath190 ( in any way ) while also adding its inflow and outflow @xmath218 . we may get , for example @xmath219 intuition suggests that if the rates of the added inflow - outflow reaction are large enough then this reaction effectively holds @xmath190 constant and thus the net effect is only to modify the rates of the reactions in which @xmath190 participates . mpne should survive under mild assumptions on the kinetics . indeed this is the case , and it is easy to make the intuitive argument precise ( theorem  [ thmnewwithopen ] ) . @xmath165    * modification 11 : `` splitting a reaction '' and inserting a complex involving a new species often preserves mpne . * suppose that we modify an existing irreversible reaction of @xmath147 by adding a new complex containing some new species @xmath190 , as an intermediate . for example we might get : @xmath220 the evolution now occurs in @xmath214 , and the stoichiometric subspace is now @xmath216-dimensional . it seems plausible that choosing the rate constants of the new reactions appropriately should lead to mpne in the new crn , and indeed this turns out to be the case , as a consequence of one of our general results ( theorem  [ thmintermediates ] ) . here , as in modification 9 , the general result allows adding several new species in several intermediate complexes provided a nondegeneracy condition is met . * modification 12 : splitting a reaction and inserting a complex involving no new species may destroy mpne . * suppose that we modify an existing irreversible reaction of @xmath147 by adding a new complex comprised of existing species as an intermediate . for example we might get : @xmath221 the evolution occurs in @xmath189 , and the stoichiometric subspace is now @xmath216-dimensional . equilibria occur when @xmath222 , and we easily compute that the only positive solution to these equations is @xmath223{\frac{k_3 ^ 2}{k_1k_2 } } , y=\sqrt[3]{\frac{k_3k_2}{k_1 ^ 2}}$ ] . thus the capacity for mpne is destroyed by this modification .      1 . if we add new `` dependent '' reactions , as in @xmath224 , then we can quite generally expect mpne and mpse to survive ( theorem  [ thmnewdepreac ] ) . this is also the subject of theorem  3.1 in @xcite . if we add new `` independent '' reactions on existing species , as in modifications 25 , then the consequences are unclear . mpne and mpse are preserved in the important and well known special case of taking the `` fully open extension '' of the original crn as in @xmath225 ( theorem  [ thmopenextension ] ) . but in general , such modifications may : * lead to the destruction of mpne as in @xmath226 . * destroy mpne while still allowing multiple _ degenerate _ equilibria as in @xmath227 . * preserve mpne as in @xmath228 . if we add species into reactions , without adding new reactions , then mpne and mpse automatically survive if the added species figure trivially in reactions ( theorem  [ thmtrivial ] ) . on the other hand , @xmath229 shows that in general we can not hope for mpne to survive , and where it does , as in @xmath230 , there may be no obvious local argument for this . if we add new species and new reactions to a crn , then mpne and mpse survive in the important special case where we add an inflow and outflow reaction for each new species , illustrated in @xmath231 ( theorem  [ thmnewwithopen ] below , and see also theorem  4.2 in @xcite ) . if we add new reversible reactions involving some new species , without adding the new species into any existing reactions ( as in @xmath232 ) , then provided certain conditions are met mpne and mpse survive ( theorem  [ thmblockadd ] ) . finally , @xmath233 and @xmath148 provide examples apparently of a rather different kind , where we modify a network by adding intermediate complexes into reactions , effectively both deleting and adding reactions . we will see that actually this modification is formally similar to adding new reactions . under certain conditions , this preserves mpne and mpse , as in @xmath233 ( theorem  [ thmintermediates ] ) . the addition of intermediate complexes is also the subject of @xcite , and we comment later on the relationship with results in this paper . we gather together the results about the inheritance of mpne , presenting the proofs later . theorems  [ thmnewdepreac]-[thmnewwithopen ] are easy and some elements of these results are well - known . theorems  [ thmblockadd ]  and  [ thmintermediates ] are harder .    in each case , @xmath3 is an unknown crn which admits mpne , and we assume that the kinetics is fixed such that mpne occurs . @xmath62 and @xmath161 are a pair of positive , compatible , nondegenerate equilibria of @xmath3 . the species of @xmath3 are @xmath17 , with corresponding concentration vector @xmath234 . the stoichiometric matrix of @xmath3 , @xmath20 , is an @xmath22 matrix with rank @xmath21 , and the reaction rates of @xmath3 are fixed at @xmath235 . columns of the matrix @xmath87 are a basis for @xmath236 , and @xmath88 is defined via @xmath89 ( see definition  [ defreddet ] ) . @xmath237 refers to the coset of @xmath67 containing @xmath62 , and by assumption @xmath238 . @xmath5 is created from @xmath3 by adding species and/or reactions , and/or modifying reactions . any new species are termed @xmath47 with concentration vector @xmath201 . in every case the new kinetics is chosen in such a way that if @xmath3 has ma kinetics , then so does @xmath5 , while if @xmath3 has positive general kinetics , then so does @xmath5 . the proofs make clear how the kinetic assumptions can be loosened , and we present remarks after the proof of each result on this .    theorem  [ thmnewdepreac ] is closely related to theorem  3.1 in @xcite . [ thmnewdepreac ] suppose we create @xmath5 from @xmath3 , by adding to @xmath3 a new irreversible reaction with reaction vector @xmath239 ( namely , @xmath240 is a linear combination of reaction vectors of @xmath3 ) . then @xmath5 admits mpne . if @xmath3 admits mpse , then @xmath5 admits mpse . [ newdeprev ] from theorem  [ thmnewdepreac ] , we could add the reverse of any existing reaction to @xmath3 as this would be a dependent reaction . equally , we could add a new _ reversible _ dependent reaction to @xmath3 and preserve mpne , as this amounts to sequentially adding two new irreversible reactions to @xmath3 . the claim about mpne in theorem  [ thmopenextension ] is well known . it appears as theorem  2 in @xcite proved using an approach similar to the one here , and as lemma  b1 in @xcite proved using techniques involving the invariance of brouwer degree , and closely related to the proof of proposition  1 in @xcite . [ thmopenextension ] suppose we create @xmath5 from @xmath3 , by adding to @xmath3 all the reactions @xmath55 for each @xmath24 in @xmath241 . then @xmath5 admits mpne . if @xmath3 admits mpse , then @xmath5 admits mpse . the following theorem is very easy to prove , and yet surprisingly important in applications when combined with other results in this paper . [ thmtrivial ] suppose we create @xmath5 from @xmath3 , by adding into the reactions of @xmath3 a new species @xmath47 which occurs with the same stoichiometry on both sides of each reaction in which it participates . then @xmath5 admits mpne . if @xmath3 admits mpse , then @xmath5 admits mpse . the following theorem is closely related to theorem  4.2 in @xcite :    [ thmnewwithopen ] suppose we create @xmath5 from @xmath3 , by adding into the reactions of @xmath3 the new species @xmath47 in an arbitrary way , while also adding the new reaction @xmath242 . then @xmath5 admits mpne . if @xmath3 admits mpse , then @xmath5 admits mpse . theorem  [ thmnewwithopen ] , combined with theorem  [ thmnewdepreac ] allows us to deduce an important corollary whose claim about mpne appears , although with different terminology , in @xcite . [ coropeninduced ] let @xmath50 and @xmath51 be fully open crns with @xmath50 an induced subnetwork of @xmath51 ( definition  [ defiso ] ) . if @xmath50 admits mpne then @xmath51 admits mpne . if @xmath50 admits mpse then @xmath51 admits mpse . we can construct @xmath53 from @xmath52 via a sequence of steps as follows : ( i ) first , for each absent species ( if any ) we add the species vertex , its inflow and outflow reaction vertices , and all missing arcs ( corresponding to theorem  [ thmnewwithopen ] ) ; ( ii ) then , for each remaining absent reaction ( if any ) we add the reaction vertex and all its incident arcs ( corresponding to theorem  [ thmnewdepreac ] because a fully open crn has stoichiometric subspace which is the whole ambient space , and hence each added reaction is now a dependent reaction ) . if @xmath50 admits mpne ( resp . , mpse ) , then since each step preserves this property , @xmath51 admits mpne ( resp . , mpse ) . the claim about mpne in corollary  [ coropeninduced ] appears as corollary  4.6 of @xcite . the claim about mpse is part of theorem  4.5 in the later work @xcite . however , it appears that our approach allows considerably weaker kinetic assumptions than needed for corollary  4.6 of @xcite . although the proofs of theorems  [ thmnewdepreac ]  and  [ thmnewwithopen ] on which corollary  [ coropeninduced ] depends are presented only for mass action kinetics or positive general kinetics , the results hold for a wide variety of kinetic assumptions as discussed in remarks  [ remnewdepreac ]  and  [ remnewwithopen ] below including , almost immediately from the proofs , any fixed power - law kinetics . joshi ( theorem 4.13 of @xcite ) showed that the fully open crns admitting mpne , minimal w.r.t . the induced subnetwork partial order , and with only one non - flow reaction are the fully open extensions of either @xmath243 or @xmath244 , where @xmath245 , respectively @xmath246 and @xmath247 . corollary  [ coropeninduced ] can be improved using theorem  [ thmintermediates ] below as indicated in remark  [ remfobest ] . the next two theorems are somewhat harder than theorems  [ thmnewdepreac ]  to  [ thmnewwithopen ] , and can be regarded as the main results of this paper . [ thmblockadd ] suppose we create @xmath5 from @xmath3 , by adding @xmath248 new reversible reactions involving @xmath68 new species with the following nondegeneracy condition : the submatrix of the stoichiometric matrix of @xmath5 corresponding to the new species has rank @xmath249 ( this implies @xmath250 ) . then @xmath5 admits mpne . if @xmath3 admits mpse , then @xmath5 admits mpse . [ thmintermediates ] suppose we create @xmath5 from @xmath3 by replacing each of the @xmath249 reactions : @xmath251 here @xmath47 is a list of @xmath68 new species which enter nondegenerately into @xmath5 in the sense that @xmath252 has rank @xmath249 ( this implies @xmath250 ) . @xmath253 , @xmath254 and @xmath255 are arbitrary nonnegative vectors and any or all may coincide . then @xmath5 admits mpne . if @xmath3 admits mpse , then @xmath5 admits mpse .    in order to understand the scope of theorem  [ thmintermediates ] , we note first that we may always formally write a single reaction as multiple reactions ; for example we way consider the single reaction @xmath256 as a set of @xmath249 such reactions , each with rate @xmath257 times the original rate . this clearly does not affect the associated dynamics and can be done while remaining in any reasonable class of kinetics ( most particularly within mass action kinetics , where each split reaction has rate constant which is the original rate constant divided by @xmath249 ) . thus theorem  [ thmintermediates ] may be used to `` split '' fewer than @xmath249 reactions , with some reactions being split multiple times and acquiring multiple intermediates . since there are no restrictions on @xmath253 and @xmath254 , many ( but not all ) instances of theorem  [ thmblockadd ] also follow from theorem  [ thmintermediates ] . for example , adding to a crn a trivial reaction with the same complex on both sides ( with any kinetics ) does not alter the vector field . given a crn with mpne , we could carry out this step first , and then `` split '' the trivial reaction using theorem  [ thmintermediates ] to create a new reversible reaction involving some new species . the resulting crn would have mpne by theorem  [ thmintermediates ] . theorem  [ thmintermediates ] appears to generalise certain aspects of results in @xcite , while being unable to reproduce others . in particular , we allow the introduction of complexes ( possibly involving existing species ) rather than just new individual species as intermediates ; however we forbid the introduction of intermediates in a degenerate way allowed in @xcite ( although see the remarks in the conclusions ) . this divergence is not entirely surprising as the techniques in @xcite are global and algebraic , while those here are local and analytical . theorems  [ thmnewdepreac ]  to  [ thmintermediates ] collectively define a partial order @xmath6 on the set of all crns : @xmath258 if and only if @xmath51 can be obtained from @xmath50 by a sequence of modifications ( possibly empty ) as described in theorems  [ thmnewdepreac ]  to  [ thmintermediates ] . reflexivity and transitivity are trivial ; antisymmetry is ensured by the fact that each modification either increases the number of species or the number of reactions or both . so , for example , the crns @xmath50 , @xmath259 , @xmath192 , @xmath215 , @xmath231 and @xmath233 in section  [ secextended ] are all @xmath260 @xmath147 . [ remfobest ] theorem  [ thmintermediates ] allows an improvement on corollary  [ coropeninduced ] . given a fully open crn @xmath3 admitting mpne ( resp . , mpse ) , we may `` split '' some reactions as in theorem  [ thmintermediates ] to get a crn @xmath5 admitting mpne ( resp . @xmath5 will have some new species , say @xmath47 , but the stoichiometric matrix of @xmath5 has full rank ( namely rank equal to the number of species in @xmath5 ) by the rank condition in theorem  [ thmintermediates ] . we may now add inflows and outflows of the species @xmath47 to get the fully open extension @xmath261 of @xmath5 . by theorem  [ thmnewdepreac ] @xmath261 admits mpne ( resp . , mpse ) since the added reactions were dependent reactions . in general @xmath261 may be a minimal crn admitting mpne ( resp . , mpse ) in the induced subnetwork partial order , but by construction is not minimal w.r.t . the partial order @xmath6 here . for example , according to theorem  4.13 of @xcite , the crn @xmath262 with mass action kinetics admits mpne and consequently , by theorems  [ thmintermediates ]  and  [ thmnewdepreac ] applied in that order , @xmath263 admits mpne . it is easy to confirm however , either by direct calculation or by applying theorem  4.13 of @xcite and theorem  3.6 of @xcite , that no fully open induced subnetwork of @xmath3 admits mpne . thus @xmath3 is a minimal fully open crn admitting mpne in the induced subnetwork partial order , while it is not minimal w.r.t . the partial order @xmath6 here . the following corollary is of importance in the analysis of biological systems , and involves the operation of adding an enzymatic mechanism to a set of reactions . it will be used a number of times in the analysis of a particular system in section  [ secbioexample ] :    [ cor : addenz ] let @xmath264 and @xmath265 be new species , not involved in @xmath266 , and let @xmath267 ( @xmath268 ) . suppose we create @xmath269 from @xmath270 by replacing each of the reactions @xmath271 then if @xmath266 admits mpne , so does @xmath269 . if @xmath266 admits mpse , so does @xmath269 . we perform successive modifications that preserve the capacity for mpne and mpse :    1 . we add the trivial species e into all the reactions simultaneously ( an application of theorem  [ thmtrivial ] ) : @xmath272 2 . we add the set of intermediate species @xmath273 ( an application of theorem  [ thmintermediates ] ) : @xmath274 3 . we add the dependent reactions @xmath275 ( @xmath249 applications of theorem  [ thmnewdepreac ] ) : @xmath276    we could see step ( 2 ) as @xmath249 applications of theorem  [ thmintermediates ] or a single application of the theorem . in either case , the nondegeneracy condition of theorem  [ thmintermediates ] is easily shown to be met . this completes the proof . before turning to the proofs of the theorems , we present an example indicating how the results here intersect with published work on biologically important systems . the mitogen - activated ( mapk ) cascade is an extensively studied network occurring in various biological processes in eukaryotic cells . the model of the mapk cascade ( figure [ fig : hffeed](f ) ) was developed by huang and ferrell @xcite , who showed that the cascade may exhibit ultrasensitive behaviour , making it appropriate for switch - like processes like mitogenesis or cell - fate determination . addition of a negative feedback in the mapk cascade ( figure [ fig : hffeed](g ) ) was demonstrated numerically to cause oscillatory behaviour by kholodenko @xcite . qiao et al . @xcite later showed numerically that the huang - ferrell model without the added negative feedback can also exhibit oscillatory behaviour and bistability . bistability in the mapk cascade has also been observed numerically and discussed in earlier papers @xcite . here we show how straightforward applications of our results prove the existence of mpne and in fact mpse in the mapk cascade , with or without negative feedback . with mass action , the mapk cascade with negative feedback ( figure [ fig : hffeed](g ) ) involves 25 species and 36 reactions ; we start by proving that the much simpler network in figure [ fig : hffeed](a ) admits mpse , and then apply a sequence of modifications that preserve mpse to ultimately transform network ( a ) into ( g ) and ( f ) of figure [ fig : hffeed ] .    in what follows , networks ( a)(g ) refer to figure [ fig : hffeed ] . we use the following abbreviations in order to aid readability of the reaction networks : x = mapk , y = mkk , z = mkkk .    * ( a ) a minimal multistationary subnetwork . * the two first phosphorylation steps of x ( namely , mapk ) are induced by y - pp ( namely , mkk - pp ) , but only the first one involves an intermediate species . both dephosphorylation steps are direct and involve no enzyme . the reactions are as follows : @xmath277 this network involves 5 species and 6 irreversible reactions . by an exact calculation ( appendix  [ appbio ] ) one can prove that the network with mass action kinetics admits mpne . this can also be confirmed using crntoolbox @xcite . we also confirmed numerically that it admits mpse . moreover , it can be checked that the network can not be constructed from any simpler network which admits mpne using the modifications described in theorems  [ thmnewdepreac ]  to  [ thmintermediates ] . in this sense the network is a minimal network admitting mpne with respect to the partial order @xmath6 and we might refer to it as an `` atom of multistationarity '' borrowing the terminology of @xcite .    * ( b ) adding enzymes : the double phosphorylation network . * network ( a ) is a simplified version of the much - studied double phosphorylation - dephosphorylation cycle , whose capacity for bistable behaviour has been both demonstrated by numerical simulations , and shown analytically @xcite . its reaction network @xmath278 is obtained from that of ( [ eq : neta ] ) by adding enzymatic mechanisms as in corollary  [ cor : addenz ] : the two reactions @xmath279 and @xmath280 get replaced by the two chains @xmath281 and @xmath282 respectively . it follows that ( b ) , with mass action kinetics , admits mpse .    * ( c ) cascading double phosphorylations i. * a simple version of the mkk double phosphorylation , not mediated by an enzymatic mechanism , is coupled upstream of the double phosphorylation mechanism of mapk from ( b ) . the network is composed of the reactions in ( [ eq : netb ] ) , plus @xmath283 adding these two reversible reactions involving the two new species y - p ( mkk - p ) and y ( mkk ) preserves the capacity for mpse of the new network . this can be regarded as two applications of theorem  [ thmblockadd ] , or a single application of the theorem ; the nondegeneracy condition is easily seen to be satisfied . therefore ( c ) , with mass action kinetics , admits mpse .    * ( d ) cascading double phosphorylations ii . * here the upstream double phosphorylation in ( c ) is modified to include enzymatic mechanisms . the reaction network of ( d ) @xmath284 is obtained from that of ( c ) by replacing the four irreversible reactions ( [ eq : netc ] ) by the four chains @xmath285 corollary  [ cor : addenz ] guarantees that ( d ) admits mpse .    * ( e ) a three layer cascade . * in network ( e ) a third layer is added upstream of cascade ( d ) . the network is composed of all reactions in ( [ eq : netd ] ) and @xmath286 adding this reversible reaction involving the new species z ( namely , mkkk ) falls immediately under the scope of theorem  [ thmblockadd ] . therefore ( e ) , with mass action kinetics , admits mpse .    * ( f ) mapk cascade . * now we modify ( e ) by adding enzymatic mechanisms ( corollary  [ cor : addenz ] ) to the reversible reaction ( [ eq : nete ] ) , which is replaced by the two chains @xmath287 and @xmath288 it follows that the new network ( huang - ferrell mapk cascade @xcite , reactions below ) , with mass action kinetics , admits mpse : @xmath289    * ( g ) mapk cascade with negative feedback . * finally , we add a negative feedback loop from x - pp as in @xcite . in that paper , the model describes the feedback as noncompetitive inhibition of z phosphorylation , whose mechanism can be unpacked as ( see for example @xcite ) : @xmath290 these reactions , together with those of ( [ eq : netf ] ) make up network ( g ) . first we add the reversible reactions ( [ eq : nf1 ] ) and ( [ eq : nf2 ] ) , involving the new species e@xmath291x - pp and e@xmath291x - pp  z , to the network ( [ eq : netf ] ) . the nondegeneracy condition is easily checked , and by theorem  [ thmblockadd ] the resulting network @xmath292 admits mpse . next , let @xmath293 and @xmath294 denote the reaction vectors of ( [ eq : nf1 ] ) , ( [ eq : nf2 ] ) , ( [ eq : nf3 ] ) and e@xmath291+z @xmath295 @xmath296 , and note that @xmath297 making ( [ eq : nf3 ] ) a dependent reaction for @xmath292 . by theorem  [ thmnewdepreac ] ( and remark  [ newdeprev ] ) , we may add ( [ eq : nf3 ] ) to @xmath292 , and thus the resulting network ( g ) , with mass action kinetics , admits mpse . some additional notation and definitions simplify the presentation .    given two matrices @xmath94 and @xmath298 with the same dimensions , @xmath299 will refer to the entrywise ( or hadamard ) product of @xmath94 and @xmath298 , namely @xmath300 . when we apply functions such as @xmath301 and @xmath302 with a vector or matrix argument , we mean entrywise application . similarly , if @xmath234 and @xmath303 , then @xmath304 . we write @xmath305 for @xmath306 . we denote the cardinality of a finite set @xmath67 by @xmath307 . we write @xmath308 for a vector of ones whose length is inferred from the context , and given some parameter @xmath116 , we write @xmath309 for @xmath310 .    given @xmath311 and @xmath312 , @xmath313 is an abbreviation for the monomial @xmath314 . if @xmath94 is an @xmath315 matrix with rows @xmath316 , then @xmath317 means the vector of monomials @xmath318 . the following examples demonstrate how entrywise and monomial notation greatly abbreviates otherwise lengthy calculations . let @xmath319 , @xmath320 and @xmath321 . let @xmath322 refer to the @xmath323 matrix of zeros . then ( i ) @xmath324 , ( ii ) @xmath325 , and ( iii ) @xmath326 . suppose @xmath319 , @xmath327 and @xmath328 . if @xmath329 , then @xmath330 . suppose @xmath331 , @xmath15 , @xmath332 . let @xmath333 be defined by @xmath334 . then @xmath335 . this expression is , of course , very familiar as the jacobian matrix of a rate function of a mass action crn ( @xcite for example . ) given an @xmath323 matrix @xmath94 and ( nonempty ) sets @xmath336 and @xmath337 , define @xmath338 to be the submatrix of @xmath94 with rows from @xmath240 and columns from @xmath339 . if @xmath340 , then @xmath341 : = \mathrm{det}(a(\alpha|\beta))$ ] . @xmath342 is shorthand for @xmath343 , and @xmath344 $ ] is shorthand for the principal minor @xmath345 $ ] . [ defcauchybinet ] given @xmath346 and @xmath347 , and any nonempty @xmath348 , @xmath349 with @xmath340 : @xmath350 = \sum_{\substack{\gamma \subseteq \langle m \rangle\\ |\gamma| = |\alpha|}}a[\alpha|\gamma]b[\gamma|\beta].\ ] ]    see @xcite , for example . [ lemschur ] suppose we have a matrix partitioned into four blocks @xmath351 with @xmath94 and @xmath352 square , and @xmath94 nonsingular . then @xmath353    this useful and well - known formula , based on schur complements , is presented as formula ( i ) on p46 of @xcite . [ lemhurwitz ] let @xmath354 for some @xmath355 and suppose we have an @xmath116-dependent family of matrices partitioned into four blocks @xmath356 with @xmath94 and @xmath352 square . suppose further that : ( i ) @xmath357 is defined and hurwitz ; ( ii ) @xmath358 , @xmath359 and @xmath360 are zero matrices ; ( iii ) @xmath361 is defined and hurwitz . then there exists @xmath362 s.t . @xmath363 is hurwitz for all @xmath364 . let @xmath94 be @xmath365 and @xmath352 be @xmath366 . we calculate @xmath367 by the continuous dependence of eigenvalues of a matrix on its entries , ( [ eqspeca ] ) tells us that @xmath249 eigenvalues of @xmath368 approach the eigenvalues of @xmath369 , and so there exist @xmath370 and @xmath114 s.t . for @xmath371 @xmath249 eigenvalues of @xmath368 lie in @xmath372 , where @xmath373 . on the other hand , expanding @xmath374 along the top @xmath16 rows gives : @xmath375 where @xmath376 . namely , eigenvalues of @xmath363 are the zeros of @xmath377 . but @xmath378 , so ( since @xmath379 ) the zeros of @xmath380 are the eigenvalues of @xmath381 . consider any contour @xmath153 in @xmath95 with the spectrum of @xmath381 in its interior . choose @xmath382 such that @xmath383 implies that @xmath384 on @xmath153 , and @xmath385 . then by rouch s theorem ( thm . 7.7 in @xcite for example ) , @xmath386 has the same number of zeros in @xmath153 as @xmath380 . thus ( [ eqspecb ] ) tells us that , for @xmath383 , @xmath16 eigenvalues of @xmath363 lie in @xmath153 and hence in @xmath95 ( in fact , it is easy to refine the argument to see that they approach the spectrum of @xmath381 as @xmath387 ) . the corresponding @xmath16 eigenvalues of @xmath368 lie in @xmath388 . in summary , for @xmath389 , @xmath368 has @xmath249 eigenvalues in @xmath372 and @xmath16 eigenvalues in @xmath388 , and so is hurwitz stable . the same clearly holds for @xmath363 .    we begin with the relatively easy cases where the results follow to a large extent from straightforward application of the ift or simple calculation , and only then prove the harder theorems  [ thmblockadd ]  and  [ thmintermediates ] . @xmath165    theorem  [ thmnewdepreac ] let the new reaction be @xmath390 so the new stoichiometric matrix is @xmath391 $ ] where @xmath392 . define @xmath393 by @xmath394 . we give the new reaction mass action kinetics and set its rate constant to be @xmath116 , so that the evolution of @xmath5 is governed by : @xmath395 the reduced jacobian matrix of @xmath74 w.r.t . @xmath87 is @xmath396 so @xmath397 which is nonsingular as @xmath62 is a nondegenerate equilibrium of @xmath3 . consequently , by corollary  [ corift ] , there exists , for each sufficiently small @xmath398 , an equilibrium @xmath399 on @xmath400 and such that @xmath401 . by continuity of @xmath402 , @xmath399 is nondegenerate for sufficiently small @xmath398 , and linearly stable for @xmath5 if @xmath62 was linearly stable for @xmath3 . the same arguments apply to give a nondegenerate equilibrium @xmath403 on @xmath400 and such that @xmath404 . as @xmath399 and @xmath399 are distinct for small @xmath405 , @xmath5 displays mpne , and mpse if @xmath62 and @xmath161 were linearly stable . [ remnewdepreac ] it is clear from the proof that the added reaction in theorem  [ thmnewdepreac ] can be given kinetics in any class provided the rate is a @xmath28 function of the concentrations and can be made arbitrarily small on any compact set in @xmath11 . the theorem generalises to other structurally stable objects , notably hyperbolic periodic orbits . theorem  [ thmopenextension ] the stoichiometric matrix of @xmath5 can be taken to be @xmath406 $ ] where @xmath407 is the @xmath365 identity matrix . let @xmath408 be any point on @xmath400 , and let the @xmath24th inflow - outflow reaction ( treated as a single reversible reaction ) have mass action kinetics with forward and backwards rate constants @xmath409 and @xmath116 respectively , so that @xmath410 is the rate of the @xmath24th added reaction . evolution of @xmath5 is then governed by the odes @xmath411 . let @xmath412 $ ] where the columns of @xmath413 are any basis for @xmath414 . @xmath415 is nonsingular and hence its columns are a basis for @xmath9 . define @xmath416 by @xmath417 . define @xmath418 and @xmath419 by @xmath420 and @xmath421 . note that @xmath422 . in this coordinate system equilibria of @xmath5 occur when @xmath423 ( as @xmath415 is nonsingular ) . for @xmath398 , @xmath74 has the same zeros as @xmath424 moreover @xmath425 ( as @xmath426 at @xmath427 , @xmath428 , and @xmath429 ) . differentiating gives @xmath430 which is nonsingular as @xmath431 is . by the ift there exists , for sufficiently small @xmath398 , a @xmath28 curve @xmath432 of zeros of @xmath138 satisfying @xmath433 . the same argument holds to give a @xmath28 curve @xmath434 of zeros of @xmath138 satisfying @xmath435 . by continuity of the determinant , they are nondegenerate zeros of @xmath138 for sufficiently small @xmath405 , and the same holds for @xmath74 as a quick calculation reveals that @xmath436 . since @xmath437 , @xmath438 and @xmath439 are positive and distinct for sufficiently small @xmath116 . thus @xmath5 admits mpne . the spectrum of @xmath440 is precisely @xmath441 with multiplicity @xmath442 with the remaining @xmath21 eigenvalues approaching the spectrum of @xmath443 as @xmath387 . thus if @xmath62 is a linearly stable equilibrium of @xmath3 , then @xmath399 is a linearly stable equilibrium of @xmath5 for sufficiently small @xmath405 , and the same holds for @xmath161 . in other words , if @xmath3 admits mpse , then so does @xmath5 . inflows and outflows were chosen to guarantee that @xmath400 remained invariant for @xmath5 and that for sufficiently small @xmath398 , the vector field of @xmath5 restricted to @xmath400 was @xmath116-close to that of @xmath3 restricted to @xmath400 in a natural sense . nondegeneracy of @xmath62 and @xmath161 for @xmath3 ensured that equilibria @xmath399 and @xmath403 persisted on @xmath400 for sufficiently small @xmath444 and remained nondegenerate w.r.t . moreover for @xmath398 the asymptotic stability of @xmath400 guaranteed that the equilibria @xmath399 and @xmath403 were also nondegenerate w.r.t . directions transverse to @xmath400 .    theorem  [ thmtrivial ] let @xmath5 have reaction rates @xmath445 . choose @xmath41 so that @xmath446 where @xmath447 is the stoichiometry of @xmath47 on the left of reaction @xmath25 ( assumed , w.l.o.g . , to be irreversible ) . we see that @xmath448 are the stoichiometric matrix of @xmath5 and a matrix whose columns are a basis for the stoichiometric subspace of @xmath5 respectively . @xmath449 and the evolution of @xmath5 is governed by @xmath450 since @xmath451 and @xmath452 , @xmath453 . so @xmath454 is a positive equilibrium of @xmath5 , and likewise for @xmath455 . moreover these equilibria are compatible as @xmath456 . nondegeneracy is easy to confirm as @xmath457 recall that nondegeneracy of @xmath62 for @xmath3 is equivalent to nonsingularity of @xmath431 , and so @xmath454 is a nondegenerate equilibrium of @xmath5 . a similar argument holds for @xmath455 , and so @xmath5 displays mpne . if @xmath62 is linearly stable w.r.t . @xmath57 then , by ( [ eqtriv ] ) , @xmath454 is linearly stable w.r.t . @xmath458 , and the same holds for @xmath161 and @xmath455 . thus if @xmath3 admits mpse , then so does @xmath5 . [ remkintriv ] in theorem  [ thmtrivial ] , the vector field of @xmath5 restricted to @xmath459 is precisely that of @xmath3 , and so not just mpne , but all dynamical behaviours of @xmath3 are reproduced on this invariant set . although this can be achieved naturally with mass action kinetics , it can also be achieved in other classes of kinetics where we can ensure that @xmath460 , for example any fixed power - law kinetics ( definition  4.8 in @xcite ) , or indeed various michaelis - menten or other hill - type kinetics .    theorem  [ thmnewwithopen ] let @xmath461 be the net stoichiometry change of @xmath47 in the @xmath24th reaction and @xmath462 . give the new reaction @xmath242 mass action kinetics with both rate constants set to be @xmath463 . the new rates @xmath41 for the other reactions are set as in theorem  [ thmtrivial ] ( consistent with , but not assuming , mass action kinetics for @xmath3 ) , so that @xmath445 satisfies @xmath460 ( and @xmath464 ) . define @xmath465 @xmath466 is the stoichiometric matrix of @xmath5 . let @xmath467 . @xmath5 gives rise to the following singularly perturbed system : @xmath468 for @xmath398 , @xmath469 has the same zeros as : @xmath470 moreover @xmath471 and we can calculate : @xmath472 using the fact that @xmath473 , we calculate @xmath474 . thus corollary  [ corift ] gives , for sufficiently small @xmath398 , a @xmath28 curve @xmath475 of zeros of @xmath138 on @xmath476 , such that @xmath477 . positivity of @xmath399 for sufficiently small @xmath116 is immediate as @xmath478 . nondegeneracy of @xmath399 for sufficiently small @xmath398 follows from continuity of @xmath479 and the fact that @xmath480 . in a similar way we have , for sufficiently small @xmath398 , a curve @xmath403 of nondegenerate positive equilibria of @xmath5 on @xmath481 , and such that @xmath482 . as @xmath483 , @xmath399 and @xmath403 are compatible equilibria of @xmath5 . thus @xmath5 admits mpne .    if @xmath62 is linearly stable for @xmath3 , then @xmath399 is linearly stable for @xmath5 for sufficiently small @xmath398 . define @xmath484 we can check quickly that if @xmath62 is linearly stable for @xmath3 , namely @xmath485 is hurwitz , then the hypotheses of lemma  [ lemhurwitz ] are satisfied , and for sufficiently small @xmath444 all eigenvalues of @xmath363 lie in @xmath486 . a similar argument applies to @xmath403 , and so if @xmath3 admits mpse , then so does @xmath5 . [ remnewwithopen ] kinetics of the non - flow reactions in theorem  [ thmnewwithopen ] can be chosen from various families other than mass action including any fixed power - law kinetics as described in remark  [ remkintriv ] . indeed provided that fixing @xmath487 ( and perhaps choosing new constants ) gives back precisely the kinetics of @xmath3 , the working easily carries through . the main conclusions of theorem  [ thmnewwithopen ] survive if we replace a positive equilibrium which is nondegenerate w.r.t . its stoichiometric class by a positive periodic orbit which is hyperbolic w.r.t . its stoichiometric class . the generalisation is nontrivial , however , requiring results from geometric singular perturbation theory @xcite . our next toy example illustrates why the proofs of theorems  [ thmblockadd ]  and  [ thmintermediates ] can appear rather involved . consider first a crn @xmath3 on a single species @xmath46 , with concentration @xmath73 and mass action rate constants shown : @xmath488 the evolution is governed by the differential equation @xmath489 and we immediately see that @xmath3 has equilibria at @xmath490 and @xmath491 . the stoichiometric subspace is @xmath492 , and nondegeneracy simply means that the derivative of @xmath493 is nonvanishing at @xmath490 and @xmath491 , which is immediate . so in fact @xmath3 admits mpne . now consider adding in a new species @xmath47 ( with concentration @xmath201 ) and the reaction @xmath494 and setting forward and backward rate constants for the added reaction to be @xmath495 and @xmath43 to get the new crn @xmath5 : @xmath496 with dynamics : @xmath497 the stoichiometric subspace of @xmath5 is @xmath189 , and nondegeneracy of an equilibrium of @xmath5 just means that @xmath498 evaluated at the equilibrium is nonsingular . intuition suggests that the modified crn might admit mpne : setting @xmath116 to @xmath33 gives two equilibria at @xmath499 and @xmath500 which we hope will persist for small @xmath398 . however , @xmath501 and as both @xmath502 and @xmath503 are singular , the ift can not be used directly to construct families of nondegenerate positive equilibria @xmath504 as @xmath387 and @xmath505 as @xmath387 . these equilibria exist , however : @xmath506 and @xmath507 are nondegenerate positive equilibria of @xmath5 for @xmath398 , and so @xmath5 admits mpne .    the failure of a naive application of the ift in even such a simple example motivated the work in this paper . although in this example the explicit calculations are easy enough , we illustrate a less explicit approach which gives insight into the proofs to follow . the first step , similar to the argument in the proof of theorem  [ thmnewwithopen ] , is to define a new family of vector fields @xmath508 tangent to stoichiometry classes of @xmath5 and where , for each fixed @xmath398 , @xmath508 has the same positive zeros as @xmath469 . to this end define @xmath509 via : @xmath510 and note that @xmath511 and @xmath512 are interior points in the domain of @xmath138 satisfying @xmath513 . the fact that @xmath138 is tangent to stoichiometry classes is in this case trivial as the stoichiometric subspace is @xmath189 . we easily calculate that @xmath514 and @xmath515 are nonsingular , and so , by the ift , there exist , for sufficiently small @xmath398 , equilibria @xmath516 and @xmath517 s.t . @xmath518 and @xmath519 . by continuity , for sufficiently small @xmath405 , @xmath399 and @xmath403 are distinct , @xmath520 and @xmath521 , and so @xmath522 and @xmath523 . so the curves consist of _ positive _ equilibria for @xmath398 sufficiently small . we can check nondegeneracy of the equilibria directly by checking nonsingularity of @xmath524 and @xmath525 . @xmath526 is easily seen to be nonzero for @xmath527 in neighbourhoods of @xmath43 and @xmath216 precisely by nondegeneracy of the original positive equilibria of @xmath3 at these points . this shows that @xmath5 admits mpne . this example illustrates some ideas in the proofs of theorems  [ thmblockadd ]  and  [ thmintermediates ] to follow , but does not indicate the additional difficulties involved with adding several new reactions involving possibly more new species than new reactions , and where the stoichiometric subspace is not necessarily all of @xmath9 . we begin with a lemma useful in both proofs :    [ lemnondegen ] let @xmath20 be an @xmath22 matrix , @xmath87 be an @xmath528 matrix whose columns are a basis for @xmath57 , and @xmath88 a matrix defined by @xmath89 . let @xmath240 be an @xmath323 matrix , and @xmath339 a @xmath529 matrix with rank @xmath249 . let @xmath530 and @xmath531 be some @xmath28 functions . define @xmath532 and @xmath533 by @xmath534 clearly @xmath535 . suppose there is a @xmath28 curve @xmath536 satisfying @xmath537 for some @xmath538 , @xmath539 . define @xmath540 and @xmath541 . assume that @xmath542 , and that @xmath543 is defined .    1 . suppose that ( i ) @xmath544 , and ( ii ) @xmath545 is nonsingular . then there exists @xmath546 s.t . , for @xmath547 , @xmath548 . 2 .   suppose that ( i ) @xmath549 is hurwitz stable , and ( ii ) @xmath545 is hurwitz stable . then there exists @xmath546 s.t . , for @xmath547 , @xmath550 is hurwitz stable . define @xmath551 and note that @xmath415 has rank @xmath552 and that @xmath553 . let @xmath554 . we are interested in the reduced jacobian matrix : @xmath555 which is a representative of @xmath550 .    1 . if @xmath556 is nonsingular then , as @xmath27 is @xmath28 , there exists @xmath557 s.t . @xmath558 is nonsingular for @xmath559 . then as @xmath387 , by lemma  [ lemschur ] : @xmath560\,\mathrm{det}(m_\epsilon ) \to \mathrm{det}\,n_0\mathrm{det}\,m_0 \neq 0,\ ] ] as @xmath545 is nonsingular . consequently , there exists @xmath561 s.t . @xmath562 for @xmath563 . 2 .   if @xmath545 and @xmath431 are hurwitz stable , then the assumptions of lemma  [ lemhurwitz ] are satisfied , and @xmath564 is hurwitz stable . this completes proof of the lemma . we now prove theorems  [ thmblockadd ]  and  [ thmintermediates ] . geometrical insight into the proofs is given in appendix  [ appgeom ] . theorem  [ thmblockadd ] let the @xmath249 added reversible reactions be @xmath565 define @xmath566 , with @xmath567 , @xmath568 and @xmath569 defined similarly . define @xmath392 and @xmath570 . @xmath240 is now an @xmath323 matrix , and @xmath339 is a @xmath529 matrix with rank @xmath249 by assumption , implying @xmath250 . , namely by reordering the species of @xmath47 if necessary , let the first @xmath249 rows of @xmath339 form a nonsingular square matrix so that @xmath571 , where @xmath572 is now a nonsingular @xmath366 matrix , and @xmath573 is a @xmath574 matrix . we have @xmath575 where @xmath532 is the stoichiometric matrix of @xmath5 and @xmath415 is a basis for @xmath576 . choose mass action kinetics for the added reactions and give the @xmath25th reaction forward and backward rate constants @xmath577 and @xmath578 respectively . setting @xmath579 and @xmath580 , the dynamics of the new system takes the form @xmath581 as @xmath339 has rank @xmath249 and hence represents an injective linear transformation , positive equilibria of @xmath5 are precisely the solutions of @xmath582 , namely of @xmath583 where @xmath584 . write @xmath585 so that @xmath586 can be written @xmath587 . taking logs gives : @xmath588 we multiply through by @xmath589 which is defined by assumption , and let @xmath590 , @xmath591 . fix @xmath592 and @xmath593 where @xmath594 and @xmath595 refer to the top @xmath366 submatrices of @xmath568 and @xmath569 respectively . setting @xmath596 and exponentiating again , we end up with : @xmath597 define @xmath598 by @xmath599 . with @xmath405 fixed and rate constants chosen as above , positive equilibria of @xmath5 occur precisely at @xmath600 note that @xmath601 and @xmath602 are in the domain of @xmath138 ( taken to be that of @xmath603 ) , which is open , and that @xmath604 . we compute : @xmath605 now @xmath606 is the @xmath366 identity matrix and we easily confirm that @xmath607 and @xmath608 . so , by the nonsingularity of @xmath431 and of @xmath572 , @xmath609 corollary  [ corift ] gives , for sufficiently small @xmath405 , a @xmath28 curve @xmath610 of zeros of @xmath138 on @xmath611 such that @xmath612 . for sufficiently small @xmath398 , @xmath613 ( as @xmath614 ) and @xmath615 ( as @xmath616 ) , and consequently @xmath617 , since @xmath618 thus each @xmath399 is a positive equilibrium of @xmath5 . an identical argument replacing @xmath62 with @xmath161 gives a curve of positive equilibria @xmath403 on @xmath619 approaching @xmath620 as @xmath387 . since @xmath621 , @xmath622 and @xmath623 are compatible , and hence @xmath399 and @xmath403 are compatible . we now infer nondegeneracy of @xmath399 using lemma  [ lemnondegen](a ) . define @xmath624 define @xmath625 by @xmath626 . then , using ( [ eqyhat ] ) , we compute that @xmath627 . observe that @xmath628 set @xmath629 , @xmath630 , @xmath631 ( which , by ( [ eqyhat ] ) , is defined and nonnegative ) , and @xmath632 . note that @xmath633 . we compute that , as required in lemma  [ lemnondegen ] , @xmath634 since @xmath635 as @xmath387 and @xmath636 as @xmath637 . by ( [ eqyhat ] ) , @xmath638 = p^{-\mathbf{1}^\mathrm{t}\gamma } > 0 $ ] , while @xmath639 = 0 $ ] for @xmath640 , @xmath641 , @xmath642 . applying the cauchy - binet formula to @xmath545 gives , @xmath643\beta[\theta|\langle m \rangle]^2 = ( -1)^mc_0k'\beta[\langle m \rangle]^2\neq 0,\ ] ] and so @xmath545 is nonsingular . all the conditions of lemma  [ lemnondegen](a ) are met , and we conclude that for each sufficiently small @xmath405 , @xmath399 is a nondegenerate equilibrium of @xmath5 . an identical argument applies to @xmath403 . thus @xmath5 admits mpne .    to see inheritance of multistability observe that @xmath644 @xmath545 is hence similar to the negative semidefinite matrix @xmath645 . as @xmath545 has already been shown to be nonsingular , it must in fact be hurwitz . if @xmath62 is linearly stable for @xmath3 , namely @xmath549 is hurwitz stable , then the conditions of lemma  [ lemnondegen](b ) are met , and we conclude that for each sufficiently small @xmath116 , @xmath399 is linearly stable for @xmath5 . an identical argument applies to @xmath403 , and so if @xmath3 admits mpse , then so does @xmath5 . theorem  [ thmintermediates ] the proof is similar to that of theorem  [ thmblockadd ] . recall that we create @xmath5 from @xmath3 by replacing each of the @xmath249 reactions : @xmath646 as we are assuming that @xmath647 has rank @xmath249 , w.l.o.g . ( namely , by reordering the species of @xmath47 if necessary ) , we write @xmath571 where @xmath572 is a nonsingular @xmath366 matrix , and @xmath573 is a @xmath574 matrix . also w.l.o.g ( namely , by splitting and/or reordering the reactions of @xmath3 as necessary ) let the reaction @xmath648 figure as the @xmath24th reaction in the original crn , proceeding with rate @xmath649 , and let @xmath650 . set the rate of each new reaction @xmath651 to be @xmath649 ( this is consistent with mass action kinetics if the original reaction had mass action kinetics ) and set the second added reaction in the @xmath24th chain to have mass action kinetics with rate constant @xmath652 ( @xmath268 ) . define @xmath653 and @xmath654 . note that for @xmath14 , @xmath655 under the assumption of weak general kinetics , and in particular @xmath656 . define @xmath657 , @xmath658 , @xmath659 , @xmath660 so that @xmath553 . @xmath532 is the stoichiometric matrix of @xmath5 and the columns of @xmath415 are a basis for @xmath576 . the dynamics of @xmath5 is governed by @xmath661 thus , with the kinetic choices made so far , the net effect on the vector field of `` inserting '' the complexes @xmath662 ( @xmath663 ) is the same adding to @xmath3 @xmath249 new ( pseudo)-reactions with stoichiometric matrix @xmath664 and rate vector @xmath665 . as @xmath339 has rank @xmath249 ( and hence corresponds to an injective linear transformation ) , positive equilibria of @xmath5 are solutions of @xmath666 write @xmath585 . the condition @xmath667 reads @xmath668 . we now perform some manipulations similar to those in the proof of theorem  [ thmblockadd ] . setting @xmath669 , and defining @xmath670 , @xmath671 , and @xmath672 , we end up with @xmath673 define @xmath598 by @xmath674 . then , for @xmath398 and reaction rates chosen as above , positive equilibria of @xmath5 occur precisely at @xmath600 note that @xmath601 and @xmath602 are in the domain of @xmath138 ( taken to be that of @xmath603 ) , which is open , and that @xmath604 . we compute : @xmath675 now @xmath676 , @xmath607 , @xmath608 , and so , as @xmath572 and @xmath431 are nonsingular , @xmath677 . corollary  [ corift ] gives , for sufficiently small @xmath405 , a curve @xmath610 of zeros of @xmath138 on @xmath611 such that @xmath612 . for sufficiently small @xmath405 , @xmath613 , @xmath678 ( as @xmath679 ) and @xmath615 ( as @xmath680 ) , and consequently @xmath617 as @xmath681 thus the curve @xmath399 consists of positive equilibria of @xmath5 . an identical argument replacing @xmath62 with @xmath161 gives positive equilibria @xmath403 on @xmath619 approaching @xmath620 as @xmath387 . since @xmath621 , @xmath399 and @xmath403 are compatible . we can now infer nondegeneracy of @xmath399 and @xmath403 using lemma  [ lemnondegen](a ) . set @xmath682 define @xmath683\,,\\ b_\epsilon:=\epsilon d_yf(x(\epsilon ) , y(\epsilon ) ; \epsilon ) = -\mathrm{diag}(\underline{v}(x(\epsilon)))\beta^\mathrm{t}\mathrm{diag}(\bm{\epsilon}/y(\epsilon))\ , . \end{array}\ ] ] set @xmath684 , @xmath630 and @xmath631 ( which , by ( [ eqyhat1 ] ) , is defined and nonnegative ) . note that @xmath633 . it is clear that @xmath685 as @xmath387 as the quantity in the square bracket approaches the constant matrix @xmath686 $ ] as @xmath387 . from ( [ eqyhat1 ] ) , @xmath687 $ ] is a positive constant , while @xmath639 = 0 $ ] for @xmath640 , @xmath641 , @xmath642 . then @xmath688 , and by the cauchy - binet formula , @xmath689(\beta[\theta|\langle m \rangle])^2 \to ( -1)^mc_0k'\beta[\langle m \rangle]^2\neq 0,\ ] ] and so @xmath545 is nonsingular . all the conditions of lemma  [ lemnondegen](a ) are met , and we conclude that for each sufficiently small @xmath405 , @xmath399 is a nondegenerate equilibrium of @xmath5 . an identical argument applies to @xmath403 . almost identically to the calculation in the proof of theorem  [ thmblockadd ] , @xmath545 can be seen to be hurwitz stable . if @xmath62 is linearly stable @xmath3 , namely @xmath549 is hurwitz stable , then by lemma  [ lemnondegen](b ) , for each sufficiently small @xmath405 , @xmath399 is linearly stable with respect to @xmath458 . an identical argument applies to @xmath403 , and so if @xmath3 admits mpse , then so does @xmath5 . [ remkin ] examining the proofs of theorems  [ thmblockadd ]  and  [ thmintermediates ] we see that mass action kinetics does not really play a fundamental role . most important are the equations ( [ eqyhat ] ) and ( [ eqyhat1 ] ) , namely that we can sufficiently control the rates of the added reactions to ensure that the @xmath690 values at the new equilibria approach @xmath33 as some parameter @xmath387 . a variety of kinetic assumptions would allow this control , including other power - law kinetics ( definition  4.8 in @xcite ) . the results admit generalisation in this direction noting , however , that the full - rank conditions in the theorems would now apply both to the blocks added to the stoichiometry matrices , and to the matrices of exponents . we have begun to describe how a crn may be enlarged while maintaining the properties of having multiple positive nondegenerate equilibria ( mpne ) , or multiple positive linearly stable equilibria ( mpse ) . the modifications in theorems  [ thmnewdepreac ]  to  [ thmintermediates ] collectively define a partial order @xmath6 on the set of all crns . in the biologically motivated example of section  [ secbioexample ] we identify an `` atom of mpse '' , namely a crn which admits mpse and which is minimal w.r.t . @xmath6 , and show how this atom occurs in various published models , immediately implying mpse in these models . we suspect that often the small multistationary networks described in @xcite , minimal w.r.t . the induced subnetwork partial order , are also minimal w.r.t . @xmath6 , and hence form natural building - blocks of crns displaying mpne . this remains to be confirmed . our approach was local : the main theoretical tool used was the ift and , informally speaking , we mostly constructed @xmath5 as a _ perturbation _ of @xmath3 in some sense . the results presented in this paper certainly do not exhaust the possibilities in this approach . we have chosen not to include a number of partial or isolated results in the same vein as theorems  [ thmblockadd ]  and  [ thmintermediates ] . these theorems themselves can probably be strengthened ; for example , with some added effort , reversibility of the added reactions in theorem  [ thmblockadd ] can probably be replaced with weak reversibility of the added reactions . it also seems that insisting that the original crn should have mass action kinetics may allow modifications which our basic assumptions would not permit because it forces certain relationships to hold between reaction rates ( for example , the ratio of the rates of two reactions with the same source complex would be constant ) . further , as remarked on at several points , the results permit modification in various directions where we choose classes of kinetics other than mass action . going beyond the techniques here , it seems likely that algebraic approaches may reveal network modifications which preserve mpe or mpne , but which could not be predicted using a local approach : some of the modifications in section  [ secextended ] and the results in @xcite point in this direction . ultimately the existence of mpe for mass action systems is about the cardinality of the intersection of certain algebraic varieties : it may be that adding new species or reactions in certain ways can be proved to allow mpe , but not by a local approach . working towards the `` best '' partial order on crns from the point of view of the inheritance of mpe or mpne is a project to be continued . the evolution of @xmath691 $ ] , @xmath692 $ ] , @xmath693 $ ] , @xmath694 $ ] and @xmath695 $ ] is governed by the ode system : @xmath696 it is easy to check that the system has precisely two independent conserved quantities @xmath697 where @xmath69 and @xmath698 are positive reals . at steady state , one can easily express all variables in terms of @xmath699 : @xmath700 note that positive values of @xmath699 will automatically give rise to positive values for all variables except @xmath701 . substituting the expressions above into the expression for @xmath702 and clearing denominators yields a cubic equation in @xmath699 : @xmath703x_2 ^ 2   \\ \nonumber                                      & & + [ 2n { k_1 } { k_3 } { k_5 } { k_6}- { k_3}^2 { k_5 } { k_6}-2 { k_2 } { k_3 }   { k_5 } { k_6}-m { k_1 } { k_3 } { k_5 } { k_6}- { k_2}^2   { k_5 } { k_6}\\ \nonumber   & & + 2n { k_1 } { k_2 } { k_5 }   { k_6 }   -m { k_1 } { k_2 } { k_5 } { k_6}-m { k_1 }   { k_3}^2 { k_5}-m { k_1 } { k_2 } { k_3 }   { k_5}-m^2 { k_1 } { k_3}^2 { k_4}\\ \nonumber   & & -m^2 { k_1 }   { k_2 } { k_3 } { k_4 } ] x_2 +   k_5k_6(k_2+k_3)^2n = 0\end{aligned}\ ] ]    now we impose that the coefficients of the cubic above have alternate signs ( a necessary condition for existence of three positive roots ) , and by trial and error we identify values for parameters that indeed give rise to distinct positive roots , and which moreover make @xmath704 in the steady state expression for @xmath701 . for example , @xmath705 gives @xmath706 , and it can be checked that @xmath707 , so that @xmath708 has a positive root in each one of the intervals @xmath709 . from the steady state expression for @xmath701 we get @xmath710 where @xmath711 . since @xmath712 and @xmath713 it follows that @xmath714 on @xmath715 , and therefore @xmath714 for all roots @xmath73 of @xmath708 . therefore all three @xmath699 values lead to positive @xmath701 values . finally , for this choice of parameters the reduced determinant of the system calculated in terms of @xmath699 is equal to @xmath716 , which ensures that all three steady states are nondegenerate . with the constants chosen as above , computing the steady states numerically yields : @xmath717 it was checked numerically that the first and the third steady state are linearly stable w.r.t . their stoichiometry class ( whereas the second is not ) and thus this crn admits mpse . we present a geometrical interpretation of theorems  [ thmblockadd ]  and  [ thmintermediates ] , which we hope can in future be used to use to generate further theorems in these directions . for simplicity we treat only the case of precisely @xmath249 new species entering nondegenerately into @xmath249 new reactions ( theorem  [ thmblockadd ] ) or @xmath249 new chains ( theorem  [ thmintermediates ] ) , although the ideas extend , with some additional effort , to the case where the number of new species is greater than the number of new reactions . [ deftransverse ] let @xmath718 be open . a @xmath28 map @xmath719 is _ transverse _ to a smooth manifold @xmath720 if @xmath721 implies that @xmath722 . we write @xmath723 . if @xmath724 , we say that @xmath708 is transverse to @xmath69 along @xmath725 , denoted @xmath726 , if @xmath727 implies that @xmath722 . [ lemtransverse ] let @xmath728 be open and bounded , @xmath729 a @xmath28 immersion , and @xmath730 some coset of a linear subspace . @xmath731 and @xmath732 are some norms on @xmath9 and on linear maps from @xmath733 respectively let @xmath734 be any closed set with nonempty interior @xmath735 , and assume that @xmath736 . given @xmath398 , define @xmath737 ( @xmath738 is a neighbourhood of @xmath708 in the @xmath28 compact - open topology , which we abbreviate as the @xmath28 topology . ) assume that @xmath739 consists of @xmath740 distinct points and @xmath741 . then there exists @xmath143 s.t . @xmath742 implies that : ( i ) @xmath743 ; and ( ii ) @xmath744 consists of @xmath745 distinct points . note that @xmath725 is compact , and @xmath67 is closed . ( i ) is now a consequence of theorem  2.1 in chapter  3 of @xcite which implies that the set of @xmath28 functions from @xmath746 transverse to @xmath67 along @xmath725 is open ( and dense ) in the @xmath28 topology . \(ii ) by ( i ) choose @xmath747 s.t . @xmath748 implies that @xmath743 . by a variant of the regular value theorem ( thm . 3.3 in chapter 1 of @xcite for example ) , the facts that @xmath749 is @xmath16-dimensional in @xmath750 , that @xmath736 , and that @xmath751 is a zero dimensional manifold ( a nonempty set of isolated points ) together imply that @xmath752 . this in turn implies , using ( i ) and the same variant of the regular value theorem , that @xmath753 is a zero dimensional manifold or possibly empty for each @xmath748 . next , let @xmath754 be a linear embedding with image @xmath67 , and define @xmath755 by @xmath756 . by construction , zeros of @xmath138 are in one to one correspondence with points in @xmath757 . as @xmath757 does not intersect @xmath758 and @xmath758 is compact and @xmath67 closed @xmath759 so , for @xmath760 , @xmath761 and so @xmath762 .    let @xmath763 . consider any path @xmath764 \ni t \mapsto f_t \in n_{\epsilon_0,l}(f)$ ] with @xmath765 . define @xmath766\to \mathbb{r}^{n+m}$ ] by @xmath767 . @xmath768 is compact : that it is closed follows from continuity of @xmath769 ; as @xmath770 is continuous in both @xmath771 and @xmath42 , @xmath772 \times l\}$ ] is a bounded subset of @xmath750 and as @xmath773 is defined and continuous on @xmath67 , @xmath774 is also bounded .    for each fixed @xmath771 , @xmath775 has zeros in one - to - one correspondence with the points of @xmath776 . define @xmath777 . we show that @xmath778 is locally constant , and hence constant on [ 0,1 ] . suppose @xmath779 for some @xmath780 $ ] . \(a ) @xmath778 is locally nondecreasing . the transversality condition at some @xmath781 , namely @xmath782 , is equivalent to nonsingularity of @xmath783 . so , by the ift , there exists @xmath784 and an open neighbourhood @xmath785 of the zero set of @xmath786 such that @xmath787 $ ] implies that @xmath788 contains exactly @xmath21 zeros in @xmath789 . consequently , @xmath790 for @xmath787 $ ] . \(b ) @xmath778 is locally nonincreasing . suppose there exists a sequence @xmath791 $ ] with @xmath792 and s.t . @xmath793 for each @xmath24 . by the argument in ( a ) for each @xmath794 there exists @xmath795 s.t . as @xmath797 is compact , by passing to a subsequence if necessary , we may assume that @xmath798 . as @xmath799 ( by construction of @xmath800 ) , we must have @xmath801 . by continuity of @xmath769 , @xmath802 , contradicting @xmath779 . +   + as @xmath803 and @xmath804 is locally constant on @xmath764 $ ] , @xmath805 for all @xmath806 $ ] . since the path @xmath807 was arbitrary , @xmath808 for all @xmath742 . we return to the geometry of the proofs of theorems  [ thmblockadd ]  and  [ thmintermediates ] focussing attention on @xmath62 , a nondegenerate equilibrium of @xmath3 . most notation is as in section  [ secthms ] . we write @xmath809 for the concentrations of species in @xmath5 . let @xmath810 be the natural embedding of @xmath9 ( the state space of @xmath3 ) in @xmath750 ( the state space of @xmath5 ) and denote the images of objects under @xmath810 with an overline . so , for example , @xmath811 . let @xmath264 be the positive equilibrium set of @xmath3 , namely @xmath812 . let @xmath236 , @xmath576 and @xmath813 be the span of the new reaction vectors , namely the image of @xmath664 in theorems  [ thmblockadd ]  and  [ thmintermediates ] . recall that in both of theorems  [ thmblockadd ]  and  [ thmintermediates ] , we found that positive equilibria of @xmath5 were exactly those points satisfying @xmath814 and an equation of the form @xmath815 ( equations  [ eqyhat ]  and  [ eqyhat1 ] ) . since here we are assuming for simplicity that @xmath816 is empty , the latter equation becomes @xmath817 for some @xmath28 function @xmath818 . thus for each fixed @xmath405 , the positive equilibrium set of @xmath5 is @xmath819 observe that @xmath820 is the graph of @xmath821 , and that , by continuity of @xmath822 , @xmath823 converges in the @xmath28 topology to the zero function as @xmath387 . let @xmath824 , @xmath825 , @xmath826 , and @xmath827    1 . * claim 1 . * _ there is an open neighbourhood @xmath828 of @xmath62 s.t . @xmath829 is an @xmath442 dimensional smooth submanifold of @xmath9 intersecting @xmath400 only at @xmath62 . moreover @xmath830 is the image of a map transverse to @xmath400 . _ these statements are just the geometrical interpretation of nondegeneracy of @xmath62 using the ift . to see this , write vectors of @xmath9 in the form @xmath831 where @xmath832 and @xmath833 ( any other complementary subspace will do ) . if @xmath834 is the vector field of @xmath3 , nondegeneracy of @xmath62 is equivalent to nonsingularity of @xmath835 . writing @xmath836 , the ift tells us that there exist neighbourhoods @xmath837 of @xmath838 in @xmath839 and @xmath141 of @xmath840 in @xmath841 , and a @xmath28 function @xmath842 satisfying @xmath843 , such that for @xmath844 , @xmath845 if and only if @xmath846 . @xmath847 is then a portion of @xmath264 containing @xmath62 , and lying entirely in @xmath11 . as it is precisely the graph of @xmath111 , it intersects @xmath400 only at @xmath62 . define @xmath848 , @xmath849 , so that @xmath850 is a @xmath28 injection with image @xmath830 . @xmath851 , and @xmath852 ( as @xmath850 is an injection ) . since @xmath853 includes no nontrivial element of @xmath854 , @xmath855 namely @xmath856 . the situation in @xmath857 coordinates is illustrated schematically in figure  [ figtrans](a ) . * claim 2 . * _ @xmath858 is an @xmath859-dimensional submanifold of @xmath750 intersecting @xmath860 only at @xmath861 . moreover @xmath858 is the image of a map transverse to @xmath860 at @xmath861 . _ the only thing that needs checking is transversality : the remaining claims are immediate from claim 1 as @xmath862 , and @xmath863 ( as @xmath864 is trivial ) . define the @xmath28 injection @xmath865 by @xmath866 , namely @xmath867 . @xmath868 has image @xmath858 . note that @xmath869 and @xmath870 . by the nondegeneracy assumptions on the new reactions , @xmath871 , and @xmath872 . moreover , by embedding of the objects of claim 1 in @xmath750 , @xmath873 . thus @xmath874 proving that @xmath875 . claim 3 . * _ for small @xmath398 and near @xmath861 , @xmath820 is an @xmath442 dimensional submanifold of @xmath750 which is the image of a map @xmath876 , @xmath28 close to @xmath868 . _ define @xmath877 by @xmath878 . @xmath876 is a @xmath28 injection as @xmath868 is a @xmath28 injection and @xmath822 is @xmath28 . define @xmath879 by construction @xmath880 lies in @xmath881 and as a @xmath28 embedding of @xmath837 in @xmath750 , it is an @xmath442 dimensional @xmath28 submanifold of @xmath750 . moreover @xmath882 which converges to the zero function in the @xmath28 topology as @xmath883 . * claim 4 . * _ let @xmath725 be any closed neighbourhood of @xmath838 in @xmath837 and let @xmath884 . then for any sufficiently small @xmath405 , @xmath885 , and there exists a positive @xmath399 such that @xmath886 . @xmath399 is isolated , namely there is a neighbourhood of @xmath399 in @xmath860 which contains no other elements of @xmath820 . _ @xmath885 follows from @xmath887 and lemma  [ lemtransverse](i ) . that @xmath888 is nonempty and consists of a single point follows from lemma  [ lemtransverse](ii ) . as @xmath820 ( and hence @xmath889 ) lies in @xmath881 , @xmath399 is positive . for ( any ) fixed @xmath890 , consider the open neighbourhood @xmath891 by choosing @xmath116 small enough we can ensure that @xmath892 for @xmath893 , and so @xmath894 . then @xmath895 is a neighbourhood of @xmath399 in @xmath860 , and @xmath399 is the unique element of @xmath820 in @xmath895 , confirming that @xmath399 is isolated . the construction of @xmath399 is illustrated schematically in figure  [ figtrans](b ) . ( -0.2,0 )  ( 5.8,0 ) ; ( 1.5,0 )  ( 4.5,0 ) ; ( 1.5,0 )  ( 4.5,0 ) ; at ( 2.2,0.2 ) @xmath837 ; ( 0,-0.2 )  ( 0,4 ) ; ( 0,1 )  ( 0,3 ) ; ( 0,1 )  ( 0,3 ) ; at ( 0.2,1.5 ) @xmath141 ; at ( 6,-0.2 ) @xmath194 ; at ( -0.2,4.2 ) @xmath41 ; at ( 3.3,4 ) @xmath400 ;                  ( -0.2,0 )  ( 6,0 ) ; ( 1.5,0 )  ( 4.5,0 ) ; ( 1.5,0 )  ( 4.5,0 ) ; at ( 2.4,0.2 ) @xmath900 ; at ( 1.8,1.4 ) @xmath901 ; ( -0.2,-0.2 ) ( 4,4 ) ; ( 1,1 ) ( 3,3 ) ; ( 1,1 ) ( 3,3 ) ; ( 0,-0.2 )  ( 0,5 ) ; at ( 6,-0.2 ) @xmath194 ; at ( 4,4.2 ) @xmath41 ; at ( -0.2,5 ) @xmath201 ; at ( 6.5,3.3 ) @xmath902 ;                      ( 5,2 )  ( 5,2.35 ) ; ( 6.8,2.3 ) ( 6.8,2.6 ) ; ( 5.5,1 )  ( 5.5,2.8 ) ; ( 5.5,2.8 ) ( 7.5,4.8 ) ; ( 3.5,2.8 ) ( 5.5,2.8 ) ; ( 7.7,3 ) ( 7.7,3.6 ) ; ( 7.7,4.2 ) ( 7.7,4.8 ) ; at ( 7.7,3.9 ) @xmath771 ; ( 4,1 )  ( 3.55,2.8 ) ( 5.55,4.8 ) ( 6,3 )  cycle ;    a similar argument applies to @xmath161 : we get , for sufficiently small @xmath398 , the existence of a positive equilibrium @xmath403 near to @xmath906 on @xmath907 . for sufficiently small @xmath116 , @xmath399 and @xmath403 are distinct , and as @xmath908 , they are compatible . the above arguments do not automatically give us nondegeneracy of @xmath399 and @xmath403 , but we do get that these are isolated equilibria ( on @xmath909 ) and so nondegeneracy should not be a surprise .
we study how the properties of allowing multiple positive nondegenerate equilibria ( mpne ) and multiple positive linearly stable equilibria ( mpse ) are inherited in chemical reaction networks ( crns ) . specifically , when is it that we can deduce that a crn admits mpne or mpse based on analysis of its subnetworks ? using basic techniques from analysis we are able to identify a number of situations where mpne and mpse 
 are `` inherited '' as we build up a network . 
 some of these modifications are known while others are new , but all results are proved using the same basic framework , which we believe will yield further results . 
 the results are presented primarily for mass action kinetics , although with natural , and in some cases immediate , generalisation to other classes of kinetics . 
 multiple equilibria ; chemical reaction networks    * msc . * 80a30 ; 15a15 ; 37c25