Journal of Difference Equations and Applications. Persistence in a Discrete-time, Stage-structured Epidemic Model

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1 Persistence in a Discrete-time, tage-structured Epidemic Model Journal: Manuscript D: Manuscript Type: Date ubmitted by the Author: Complete List of Authors: Keywords: GDEA R Original Article n/a alceanu, Paul; Arizona tate University, Mathematics mith, Hal; Arizona tate University, Mathematics tructured populations, R epidemic model, persistence, global stability

2 Page of xmm (00 x 00 DP)

3 uly, 00 : Persistence LJA R Page of Vol. 00, No. 00, January 00, REEARCH ARTCLE Persistence in a Discrete-time, tage-structured Epidemic Model Paul Leonard alceanu & Hal L. mith Department of Mathematics and tatistics, Arizona tate University, Tempe, AZ (v. released October 00) Discrete-time and R epidemic models, formulated by Emmert and Allen [] for the spread of a fungal disease in a structured amphibian host population, are analyzed. Criteria for persistence of the population as well as for persistence of the disease are established. Global stability results for host extinction and for the disease-free equilibrium are presented. Keywords: tructured populations, R epidemic model, persistence, global stability. ntroduction This paper is inspired by the work of Emmert and Allen [], who formulated a family of discrete-time mathematical models of a fungal infection spreading in an amphibian population. The models apply to the study of chytridiomycosis, an emerging infectious disease caused by the chytrid fungus. Thought to have originated in Africa with first global occurrence in [], the disease causes sporadic deaths in some amphibian populations and 00% mortality in others. t has been implicated in the mass die-offs and species extinction of frogs in many areas of the world such as Australia, Panama, Ecuador, Venezuela, New Zealand and pain. There is still no general consent about the impact of this disease. While some researchers say that it is the major primary cause of extinction and declines of frog populations [], others claim that the fungus indeed affects a wide variety of frog species, but extinction and declines have occurred only in a few []. ndividual frogs contact the disease either when their skin comes into contact with water that contains spores from infected animals, or by direct contact with any infected animal. After infection, the fungus invades the surface layers of the frog skin, causing damage to the keratin layer. The models described in [] incorporate the life-stage structure (larva, juvenile and adult) of the amphibian population afflicted with the fungal disease. Emmert and Allen explore various sub-models which simplify the stage structure of the host and thereby are more mathematically tractable. Their mathematical results establish persistence and extinction of the population, but persistence is established in a somewhat narrow sense: the existence of a disease-free equilibrium. Numerous numerical simulations and bifurcation diagrams show that quite complicated dynamic behavior is possible in the models, especially with the Ricker recruitment nonlinearity and large intrinsic birth rates. Our aim in this paper is to use the mathematical theory of persistence [, ] to establish persistence results for the Corresponding author. salceanu@mathpost.asu.edu N: 0- print/n -0 online c 00 Taylor & Francis DO: 0.00/0YYxxxxxxxx

4 uly, 00 : Persistence LJA R Page of full (larva, juvenile, adult) stage-structured model. This theory is most well developed when the boundary dynamics are relatively simple and so our results, while establishing persistence in a stronger sense than in [] will, in some cases, require the restrictive assumption that disease-free dynamics is convergent. We provide a crude sufficient condition for this to hold and observe from numerical simulations that it seems to hold for a much larger parameter range. From a biological perspective, we are mainly interested in two issues. Can the disease drive the population to extinction, or not? Does the disease persist in the population, i.e., is it endemic in the population? n the context of the mathematical model of Emmert and Allen, we show that the answer to the first question is no, the disease cannot drive a population which persists in the absence of disease to extinction. We provide sufficient conditions for both the eradication of the disease from the population and for its persistence. As pointed out to the authors by a referee, extinction of the amphibian host can occur, at least in numerical simulations, in more recent models of chytridiomycosis due to Briggs et al. [] and Mitchell et al. []. The model of Mitchell et al. is deterministic and amphibian extinction was made possible by setting a threshold level below which the population was set to zero. Extinction may occur in the model of Briggs et al. because it is stochastic. Thieme et al. [0] propose a traditional ordinary differential equation model with mass action incidence, but with general disease-free host dynamics that may include an Allee effect. Among many interesting outcomes, one is host eradication due to the disease. The paper is organized as follows: in the following section, two models are described, one with recovery from the disease and one where recovery does not occur. n section three, we treat the model without recovery and in section four the model with a recovery class. Results are stated without proof. ection five contains some mathematical tools used repeatedly, and the proofs of our results are given in section six. We denote the m-dimensional Euclidean space by R m and the positive cone in this space by R m + = {x R m x i 0, i =,..., m}. The interior of R m + is (R m + ) 0 = {x R m x i > 0, i =,..., m}. N denotes the set of non-negative integers, and O mn the m n zero matrix. Whenever the dimension of O mn will be clear from the context, we will omit the subscripts.. The Emmert-Allen Model The Emmert-Allen model [] considers Larvae L, Juvenile J, and Adult A stages of the host population, labeled with subscripts,, R indicating their status as susceptible, infective, or recovered. F denotes density of fungus in the environment. n addition, all parameters in the model are assumed to be positive, except c L, c L, c LR, c J, c J and c JR (see below), which are assumed to be non-negative (we allow them to be zero). nterchanging the equations, we re-write the LJA-R model as in (), where T in the density-dependent birth rates b K φ(t ), K =,, R is T = c L L + c L L + c LR L R + c J J + c J J + c JR J R + c A A + c A A + c AR A R. The function φ(x) : [0, ) [0, ) is strictly decreasing and x xφ(x) is bounded. A common choice is φ(x) = e x (Ricker), or φ(x) = /( + x) (Beverton- Holt). The probability of becoming infected in state N, e βn w, N = L, J, A, increases with increasing values of infection in the population and environment,

5 uly, 00 : Persistence LJA R Page of infection represented by the weighted sum of the infected stages and fungus: w = w L L + w J J + w A A + w F F. The model assumes a mass-action transmission rate. As pointed out in [], one may assume a transmission rate that is proportional to the population size (see []). L n+ = p L L n e β Lw n + b φ(t n )A n + b Rφ(T n )A n R J n+ = q L L n + p JJ ne βjw n A n+ = q J J n + p AA n e β Aw n L n+ = p L L n ( e βlw n ) + p L L n + b φ(t n )A n J n+ = q L L n + p JJ n( e β Jw n ) + p J J n A n+ = q J J n + p AA n ( e β Aw n ) + p A A n L n+ R = p L L n + p LRL n R J n+ R = q LR L n R + p JJ n + p JRJR n A n+ R = q JR JR n + p AA n + p ARA n R F n+ = b F (v L L n + v JJ n + v AA n ) + p F F n The following inequalities among parameters are assumed: p L + q L < p LR + q LR < p J + q J < p JR + q JR < p A < p AR < p F < p L + p L + q L < p J + p J + q J < p A + p A < Emmert and Allen [] assume that the infective birth rate is smaller than the others, (i.e., b b, b R ), but it is not used. When there is no recovered class, the model () becomes an LJA- model: now with L n+ = p L L n e βlw n + b φ(t n )A n J n+ = q L L n + p JJ ne β Jw n A n+ = q J J n + p AA n e βaw n L n+ = p L L n ( e β Lw n ) + p L L n + b φ(t n )A n J n+ = q L L n + p JJ n( e β Jw n ) + p J J n A n+ = q J J n + p AA n ( e βaw n ) + p A A n F n+ = b F (v L L n + v JJ n + v AA n ) + p F F n T = c L L + c L L + c J J + c J J + c A A + c A A. Remark. The Emmert-Allen models described above assume that vertical transmission of disease from adult to larvae is 00% efficient. This may not be the case. The models are easily modified to allow for a vertical transmission efficiency () () ()

6 uly, 00 : Persistence LJA R Page of f [0, ]. One simply replaces b φ(t n )A n by f b φ(t n )A n in the equation for L and adds ( f) b φ(t n )A n to the equation for L. Proposition.. f c AK > 0, K =,, R (K =, ) then () (respectively ()) has a global attractor of bounded sets. n particular, there is a bounded set that attracts all orbits.. Dynamics of the LJA- Model Our main results for the LJA- model are given in this section. We begin with some useful notation. Denote (L, J,..., F ) R + in short by x, and consider the matrices A K (x) = p LK 0 b K φ(t ) q LK p JK 0 ; K =,. () 0 q JK p AK Let A K := A K (0) = p LK 0 b K q LK p JK 0. () 0 q JK p AK Notice that A K is an irreducible matrix. Let à (x) be the matrix obtained from A (x) by replacing p N by p N e βn w ; N = L, J, A. Then our LJA- model (model ()) can be written as x n+ = A(x n )x n, n N, where the nonnegative matrix A(x) is of the form à (x) O O A(x) = à (x) A (x) O. O à p F Observe that Ã(x) = O, x with x = x = x = x = 0. The Jacobian at the extinction state x = 0, denoted by J(0), is the same as A(0). o, Clearly, its spectral radius, r(j(0)), satisfies Let K = A O O J(0) = O A O. () O à p F r(j(0)) = max{r(a ), r(a ), p F }. q LK q JK b K ; K =,. () ( p LK )( p JK )( p AK ) The relationship between K and r(a K ) is established in the following

7 uly, 00 : Persistence LJA R Page of Lemma.. ( K )( r(a K )) 0, with equality if and only if K =. From the form of matrix A(x) we see that we have the following positively invariants sets of (): B F = {x R + L = J = A = F = 0}, B = {x R + L = J = A = 0} and the F axis = {x R + L = J = A = L = J = A = 0}. B F represents no disease states, B represents no susceptible states (disease only), and the F axis represents no host states. From the biological point of view, the invariance of B is due to the assumed 00% efficiency of vertical transmission: infected adults produce infected eggs which become infected larvae. Following [], whenever K > we define L K = ( p JK)( p AK ) C φ ( K ) J K = q LK( p AK ) C φ ( K ) K =,, Ā K = qlkqjk C φ ( K ) where C = c LK ( p JK )( p AK ) + c JK q LK ( p AK ) + c AK q LK q JK. Let Proposition.. () F = b F (v L L + v J J + v A Ā ). () p F The nonzero boundary fixed points of () are as follows: a) E B F exists if and only if >. When it exists, E is given by (), with K =. b) E B exists if and only if >. When it exists, E is given by (), with K = and by ().. Disease-free Dynamics When there is no disease present, the model takes the form: L n+ J n+ A n+ = p L L n + b φ(t n )A n = q L L n + p JJ n = q J J n + p AA n now with T = c L L + c J J + c A A. Let x = (L, J, A ). Theorem.. The dynamics of (0) are given by: a) f < then 0 is a globally asymptotically stable fixed point. b) f > then there exists ε > 0 such that lim inf min n i xn i > ε, x 0 R + \ {0}. Part a) can be found in []. We let Q : R + R + be defined by the right hand side of (0). Also, we consider an analogous map, denoted by Q, defined by the same formula as Q, (0)

8 uly, 00 : Persistence LJA R Page of Table. Parameter values for global convergence to E K. Parameter c LK c JK c AK b K p LK p JK p AK q LK q JK A B Figure. Two orbits converging to the positive fixed point (φ(x) = e x ). but with replaced by, and for both maps (K =, ) we give the following result: Theorem.. a) φ(x) = +x and Γ Assume K >, c JK = 0, and let Γ := clkbk c AKp LK. f either { K, if K K, if K > b) φ(x) = e x, K e and Γ K ln( K), then ( L K, J K, ĀK) is an asymptotically stable fixed point of Q K and attracts all non-zero initial data. Results of [] show that the conclusion of Theorem. extends to nearby parameter values. n row A of Table we give a set of parameters satisfying all conditions in Theorem., for both choices of φ. Row B consists of parameters taken from Table in [], with b K =. For these parameters, we may still have global convergence to E K as suggested by Figure. Thus, we conjecture that global convergence of E K holds on a much larger parameter set than Theorem. suggests.. Main Results for the LJA- Model Motivated by Theorem. and the simulation suggesting that E K attracts non-zero trajectories for parameter values not satisfying the conditions of the theorem above, we will make occasional use of the following hypotheses (K =, ): (C K ): ( L K, J K, ĀK) is asymptotically stable fixed point of Q K and attracts the orbits (under Q K ) of all points in R + \ {0}. or, Remark. The following will be proved in ection : a) (C ) E is asymptotically stable in B F and attracts non-zero initial data in B F. b) (C ) E is asymptotically stable in B and attracts initial data in B \ (F axis). Using (), let be T evaluated at E K, K =,, and T K = c LK LK + c JK JK + c AK Ā K ; K =, () w Ī = w L L + w J J + w A Ā + w F F ()

9 uly, 00 : Persistence LJA R Page of be w evaluated at E, and consider the matrices p L e β Lw Ī 0 b φ( T ) J = q L p J e βjw Ī 0, () 0 q J p A e βaw Ī which represents the submatrix of the Jacobian of () (and also of ()) evaluated at E, formed with the elements situated at the intersection of rows,, with the columns,,, and 0 p L + p L β L w L L p L β L w J L p L β L w A L + b φ( T ) p L β L w F L J F = q B@ L + p J β J w L J p J + p J β J w J J p J β J w A J p J β J w F J p A β A w L Ā q J + p A β A w J Ā p A + p A β A w A Ā p A β A w F Ā CA, () b F v L b F v J b F v A p F which represents the sub-matrix of the Jacobian of () (and also of ()) evaluated at E, formed with the elements situated at the intersection of rows,,,0 with the columns,,,0. Let Ĵ F be the matrix obtained from J F by replacing φ( T ) by. Now we give a result regarding the local and global stability of the boundary fixed points; assertion b) is due to []: Theorem.. The following hold: a) f <, then L n, J n, An 0. b) 0 is asymptotically stable if <, < and unstable if either > or >. n the former case, 0 is globally asymptotically stable. c) f E is asymptotically stable in B F and r(j F ) < then E is asymptotically stable in R +. f r(ĵ F ) < and hypotheses of Theorem. hold, then L n, J n, An, F n 0 and, if, E attracts all initial data in R + \ B. d) f E is asymptotically stable in B and r(j ) <, then E is asymptotically stable in R +. Assuming < and (C ), then E attracts all initial data in R + \ (B F F -axis). ince it is not obvious that the conditions in parts c) and d) of Theorem. are feasible, we provide below two sets of parameters for which these conditions hold for both choices of φ (Ricker and Beverton-Holt): conditions in c) are satisfied with all the parameters regarding the susceptible population (K = ) as in Table row A, and all the parameters regarding the infective population (K = ) as in row B, except for b = ; conditions in d) are satisfied with all the parameters regarding the susceptible population (K = ) as in Table row B, except for b =, and with all the parameters regarding the infective population (K = ) as in row A, except for q L = 0., q J = 0., b =. Parameters not given in the table are as follows: w N = 0., v N = β N =, N = L, J, A, w F = 0., p F = 0., b F = 0. The reader should notice, in the previous result, that the conditions regarding asymptotic stability on the subspaces are part of hypotheses (C K ), K =,. The condition > is an indicator of the viability of the population: if < the population cannot survive in the absence of the disease. n case c), the hypothesis E is asymptotically stable in B F means that E is stable for the disease-free model. The spectral radius r(j F ) is a measure of the ability of the disease to invade the disease-free state. Therefore, E is stable if it is stable for the disease-free model and if the introduction of a small group of infective hosts does

10 uly, 00 : Persistence LJA R Page of not lead to an outbreak. The assumption of 00% efficient vertical transmission leads to the possibility of a susceptible-free population, unusual in epidemiology. Thus, the disease-only equilibrium E may exist, in which case has similar stability properties as E. As noted in case d), the spectral radius r(j ) measures the ability of a few susceptible host to invade E. The relationship of stability properties of E and E is unknown in general. t is possible that they have the same stability, either both stable or both unstable (see below, in the paragraph following Theorem., a set of parameters for which both E and E are unstable). Below we present the main persistence result of this section. Theorem.. Persistence and Positive Fixed Points: ()a) f > and > then there exists ε > 0 such that lim inf n Ln + J n + A n + L n + J n + A n > ε, x 0 R + \ F-axis. b) f > and < then there exists ε > 0 such that lim inf n min{ln, J n, A n } > ε, x 0 R + \ B. () Assume >, r(j F ) > and (C ). f a) > then there exists ε > 0 such that lim inf n min{ln, J n, A n, F n } > ε, x 0 R + \ (B F F axis). f, in addition, r(j ) > and (C ) holds, then there exists ε > 0 such that lim inf min n i xn i > ε, x 0 R + \ (B F B ), and () has a fixed point in (R +) 0. b) < then there exists ε > 0 such that lim inf min n i xn i > ε, x 0 R + \ (B F B ), and () has a fixed point in (R +) 0. () f < and >, then there exists ε > 0 such that lim inf n min{ln, J n, A n, F n } > ε, x 0 R + \ (B F F axis). Again, in order to check the feasibility of the conditions in part () a) of the Theorem., for both Ricker and Beverton-Holt forms of φ, we provide the following set of parameters: c LK = 0.0, c JK = 0, K =,, c A =., c A = 0.; b =, b =, b F = 0; p LK = 0.0, p JK = 0.0, p AK = 0.0, K =,, p F = 0.; q L = 0., q J = 0., q L = 0., q J = 0.; w F = 0.0, w N = 0.0; v N = β N =, N = L, J, A. As mentioned earlier, the case that the extinction fixed point is unstable is of interest since our focus is on the disease. o, we are primarily concerned with the situation when >. We have shown (Theorem. case ()) that the disease cannot drive the population to extinction. When the disease-free state is unstable, additional conditions, given in Theorem. case (), imply that the disease is

11 uly, 00 : Persistence LJA R Page 0 of endemic in the population and there exists a positive fixed point. When the diseasefree state is asymptotically stable, additional conditions given in Theorem. imply that it is globally stable, and the disease is eradicated. Case () of Theorem. is included for mathematical completeness; it seems unlikely that the host population is nonviable in the absence of disease but viable when all its members are infected.. Dynamics of the LJA-R Model Let x = (L, J,..., F ) R 0 +. The following sets are positively invariant for (): B RF = {x R 0 + L = J = A = L R = J R = A R = F = 0}, B F = {x R 0 + L = J = A = F = 0} and the F axis. B RF represents no disease and no recovery states, B F represents no disease states, and the F axis represents no hosts states. Notice that 0 is always a fixed point of (). Proposition.. The only possible non-trivial boundary fixed point of () is E = ( L, J, Ā, 0,.., 0), with L, J, Ā given in (). E exists if and only if >.. Dynamics with nfectives and Fungus removed f infectives and fungus are removed we are left with only the susceptible and recovered populations. We describe below the dynamics of the model () restricted to these two stages: L n+ J n+ A n+ L n+ R J n+ R A n+ R = p L L n + b φ(t n )A n + b Rφ(T n )A n R = q L L n + p JJ n = q J J n + p AA n = p LR L n R = q LR L n R + p JRJR n = q JR JR n + p ARA n R now with T = c L L + c J J + c A A + c LR L R + c JR J R + c AR A R. Persistence and extinction for () are treated in our next result. Theorem.. () The recovered population goes extinct, i.e. L n R, J R n, An R 0, and a) f < then 0 is a globally asymptotically stable fixed point. b) f > then there exists ε > 0 such that lim inf n min{ln, J n, An } > ε, x0 R + \ {0}. As in ection., we make the following remark, which will be proved in ection : Remark. (C ) E is asymptotically stable in B F and attracts all non-zero solutions to ().

12 uly, 00 : Persistence LJA R Page of Main Results for the LJA-R Model Now we give a result regarding the local and global stability of the boundary fixed points 0 and E ; assertion about the local asymptotic stability of 0 in a) is due to []. Let Theorem.. p L 0 b b R 0 q L p J q J p A p L 0 0 p L 0 b B = 0 p J 0 q L p J p A 0 q J p A p L 0 0 p LR p J 0 q LR p JR p A 0 q JR p AR b F v L b F v J b F v A p F The following hold: a) 0 is asymptotically stable if <, < and unstable if either > or >. f r(b) < then 0 is globally asymptotically stable. b) f E is asymptotically stable in B RF and r(j F ) <, then E is asymptotically stable in R 0 +. Easily verifiable conditions for r(b) < to hold are not easy to find. Thus, we provide an alternative, more checkable sufficient condition for 0 to be globally asymptotically stable: Remark. Let p N = max{p N, (p N + p N ), p NR }, N = L, J, A; q N = max{q N, q N, q NR }, N = L, J; and b = max{b, b, b R }. Notice that p N <, N = L, J, A (see ()). f then 0 is globally asymptotically stable. q L q J b = ( p L )( p J )( p A ) < A set of parameters for which < is: p LK = 0.0, p JK = 0.0, p AK = 0.0, K =,, R; q LK = 0., q JK = 0., K =, ; and b = 0, b R, b 0. Except for the b K s, these parameters are taken from Table in []. Now we present the main result of this section: Theorem.. Persistence and Positive Fixed Points: () f > then there exists ε > 0 such that lim inf n min{ln, J n, A n } > ε, x 0 R 0 + \ F axis. () f one of the following holds: a) >, r(j F ) >, and (C ) b) < and > then there exists ε > 0 such that lim inf n min i 0 xn i > ε, x 0 R 0 + \ (B F F axis),

13 uly, 00 : Persistence LJA R Page of and () has a fixed point in (R 0 + ) 0. As for the LJA- model, we are primarily interested in the case that the host population persists in the absence of the disease. This amounts to the assumption that >. n that case the population persists, so the disease cannot drive the host population to extinction. However, we show that the disease can become endemic, that is, persists in the host population and there is a positive fixed point, if r(j F ) > and if the disease-free dynamics is simple (i.e., converges to E ). The main difference between the LJA- and LJA-R models is that infectives have an outlet to the recovered class so there is no susceptible-free invariant subspace and the associated technicalities.. Mathematical Preliminaries First, we provide a series of definitions and introduce some notation. We denote the spectral radius of the matrix A by r(a), and the transpose of A by A T. We call the matrix A = (a ij ) i,j m R m R m positive, and write A > O, if a ij 0, i, j {,..., m}, and a kl > 0 for some k and l; strictly positive, and write A O, if a ij > 0, i, j {,..., m}; non-negative, and write, A O, if a ij 0, i, j {,..., m}. We define the partial order relation on the set of non-negative matrices as follows: A B B A O. Assume analogous definitions for < and. Also, we assume the same notation for vectors in R m. Consider the following norm on R m : m x = x i, where x i represents the absolute value of x i. i= For a differentiable function f : R m R m we denote the derivative of f at x by Df(x). The state space for the models considered here will always be R m +, for some m N, so all the sets we will work with, like stable manifolds or neighborhoods of fixed points are assumed to be so, relative to R m +. We denote sequences that come from the iteration of the difference equation in a model, by using superscripts (i.e., x 0, x, etc.). For any other sequences we use subscripts. The components of a vector x R m + are denoted by x, x,..., x m. Below we give a series of definitions related to the difference equation x n+ = f(x n ) () where x 0 R m +, and f : R m + R m + is a continuous map. The positive orbit of x 0, which hereafter we will refer to, in short, as the orbit of x 0, is {x n n 0}. We denote the omega limit set of x 0 by ω(x 0 ). Definition.. Let X R m + be a positively invariant set for (), and E X a fixed point. E is called stable in X, if for any neighborhood V of E there exists a neighborhood U of E such that x 0 U X x n V X, x 0 U, n 0. f, in addition there is a neighborhood W of E such that x n E as n for any x 0 W X, then E is called asymptotically stable in X;

14 uly, 00 : Persistence LJA R Page of globally asymptotically stable in X, if it is asymptotically stable in X and attracts all solutions starting in X. Definition.. The stable manifold of a fixed point E of () is W (E) = {x R m + x n E as n ; x 0 = x}. Definition.. A set M R m + is called invariant if f(m) = M. Definition.. A fixed point E of () is called isolated in R m +, if there exists a neighborhood of E in which there is no invariant set other than {E}. Definition.. A function ϕ : Z R m is called a total trajectory (of ()), if n N, s Z, x 0 = ϕ(s) x n = ϕ(n + s). f n 0 N, x 0 R m and ϕ(n 0 ) = x 0, then ϕ is called a total trajectory through (n 0, x 0 ). φ(z) is called the total orbit of φ. The alpha limit set of ϕ, denoted by α(ϕ), is defined in the usual way. Let ρ : R m + [0, ) be continuous and not identically zero. We will make the assumption: (H) x 0 R m +, ρ(x 0 ) > 0 ρ(x n ) > 0, n 0. n all our applications, ρ satisfies (H) and we will not explicitly verify it. Definition.. We call () uniformly strongly ρ-persistent if there exists ɛ > 0 s.t. lim inf n ρ(xn ) > ɛ, x 0 R m + such that ρ(x 0 ) > 0. We often drop the word strongly and just say ρ-persistent. and We will work with the following sets: 0 = {x R m + ρ(x n ) = 0, n 0} () Ω = x 0 ω(x). () Definition.. Let C, B 0. C is said to be chained to B in 0, written C B, if there exists a total trajectory ϕ with compact orbit closure through some x C B with ϕ(z) 0 such that ω(x) B and α(ϕ) C. A finite sequence {M,.., M k } of subsets of 0 is called cyclic if, after possibly renumbering, M M in 0, or M M.. M j M in 0 for some j {,.., k}. Otherwise it is called acyclic. The following theorem is a particular case of [, Theorem..] where, in the notation of that result, X 0 = 0. Theorem.. Assume f as in () is a continuous map, ρ as above, and the following hold: ) f has a global attractor ) There exists a finite sequence M = {M,..., M k } of disjoint, compact and invariant sets in 0 having the properties: a) Ω k i=m i b) M is acyclic c) M i is isolated in R m +, i =,..., k d) W (M i ) 0, i =,..., k Then () is uniformly ρ-persistent.

15 uly, 00 : Persistence LJA R Page of Lemma.. Let ρ be given satisfying (H), 0 as in (), and M = {E, E,..., E k } 0, where E, E,..., E k are fixed points of (). f E is asymptotically stable in 0, then there are no cycles among the members of M that contain E. Proof. We need to rule out the two possible situations that can occur: ) E E i, i =,..., k. t is clear that this situation cannot occur, because any trajectory starting arbitrarily close to E is attracted to E, so it cannot have E i as its omega limit set. ) E E. uppose there is a total trajectory ϕ through some (0, x 0 ) 0 such that α(ϕ) = ω(x 0 ) = E and whose total orbit is contained in 0. Then there exists V a neighborhood of E such that the total orbit of ϕ is not contained in V. o let s Z such that ϕ(s ) V (i). ince E is asymptotically stable in 0, we have that there exists a neighborhood U V of E such that any solution to () that starts in U stays in V at any time. ince α(ϕ) = {E}, s < s such that ϕ(s ) U. Now consider the solution to () starting at x 0 = ϕ(s ). Then x s s V. But x s s = ϕ(s ), and we have a contradiction to (i). Consider that an equation like () is written in the form { x n+ = F (z n ) y n+ = G(z n ), () where z n = (x n, y n ) R k + R m k +, and F : Rm + R k +, G : R m + R m k + are continuous maps. Then, we have the following lemma: Lemma.0. Let X 0 = {(x, y) R k + R m k + x = 0}, and E X 0 be a fixed point of (). Assume that F (z) A(z)x, z = (x, y) R m +, (0) where A(z) is a k k continuous matrix. f A(E) T has an eigenvalue greater than one, associated with a strictly positive eigenvector, then: a) There exists a neighborhood of E that does not contain any positively invariant set M (of ()) satisfying M (R m + \ X 0 ). b) f E is asymptotically stable in the largest positively invariant subset of X 0, then E is isolated in R m +. c) f R m + \X 0 is positively invariant for (), then W (E) is included in the largest positively invariant subset of X 0. and v 0 a corre- Proof. Let A 0 := A(E). Let λ > be an eigenvalue of A T 0 sponding eigenvector. From (0) we have v T x n+ λv T x n v T (A 0 A(z n ))x n, n N Let v := min{v i }, v := max{v i } and ε > 0 such that λ ε >. ince v 0, we i i have v, v > 0 and let α := v v > 0. A(z) is continuous (at E) implies that δ > 0 such that if z E < δ, then ε kα a0 ij a ij(z) ε kα, i, j {,.., k}, where a0 ij and a ij (z) denote the elements of A 0 and A(z), respectively. o, if z n E < δ, (i).

16 uly, 00 : Persistence LJA R Page of using the fact that v is positive, we obtain the following inequality: v T (A 0 A(z n ))x n which, combined with (i), yields: ε kα v xn ε kα k v xn = εv x n εv T x n, v T x n+ λv T x n εv T x n = (λ ε)v T x n = µv T x n with µ := λ ε >. a) uppose in any neighborhood of E there exists M a positively invariant set such that M (R m + \ X 0 ). o, consider M B δ (E) = {z R m + z E < δ}. Let z 0 M (R m + \ X 0 ). Then, z n B δ (E), n 0 (iii). From (ii) it follows that v T x n µ n v T x 0, n 0. But since z 0 X 0 and v T 0, we have v T x 0 > 0, hence v T x n x n as n, contradiction to (iii). b) Let be the largest positively invariant set in X 0. ince E is asymptotically stable in, there is an open neighborhood V of E in R m + such that x n E whenever x 0 V. We can choose V such that V B δ (E). uppose E is not isolated in R m +. Then V contains some invariant set M {E}. Without loss of generality we may assume M is closed. By part a) we have M. Let x 0 M \ {E}. M being invariant, there exists a total trajectory ϕ : Z M, ϕ(0) = x 0, whose alpha limit set, α(ϕ), is included in M V. Thus, any solution starting in α(ϕ) converges to E as n, and since α(ϕ) is closed and invariant, we have that E α(ϕ) (iv). Now again, E being asymptotically stable in, there exist V, V neighborhoods of E such that V V, x 0 V, and such that any solution that starts in V stays in V for any n 0. From (iv) we have that there exists N N such that ϕ( N) V. Let y 0 = ϕ( N). Then y N V. But y N = ϕ( N) = x 0, so we have a contradiction. Hence E is isolated in R m +. c) uppose V = W (E) (R m + \ X 0 ). Let U be a neighborhood of E. Then, if x 0 V we have that there exists N N such that x n U, n N. But since R m + \ X 0 is positively invariant, we have that x n R m + \ X 0, n N. Thus, {x n } n N is a positively invariant set contained in U (R m + \ X 0 ). ince U was arbitrarily chosen, we have a contradiction to a). Hence W (E) (R m + \ X 0 ) =, or equivalently, W (E) X 0. But since W (E) is itself a positively invariant set, we have, in fact, that W (E) is contained in the largest positively invariant subset of X 0. (ii), Corollary.. Assume F is C, and X 0 = {z R m + x = 0} is positively invariant for (). Let E X 0 be a fixed point of () and C(z) = ( Fi x j (z)) i,j k. Then C(z) 0 for any z X 0. f C(E) T has an eigenvalue greater than one, associated with a strictly positive eigenvector, then: a) There exists a neighborhood of E that does not contain any positively invariant set M satisfying M (R m + \ X 0 ). b) f E is asymptotically stable in X 0, then E is isolated in R m +. c) f R m + \ X 0 is positively invariant for () then W (E) X 0. Proof. First, let s show that C(z) 0 for any z X 0. o let z X 0, and z 0 = z. Then, since X 0 is positively invariant, z X 0, so x i = 0, i =,..., k, which implies 0 = x i = F i(z 0 ) = F i (z), i =,..., k. Thus, for any i, j =,..., k, we have F i x j (z) = F i (0,.., h,.., 0, y,.., y m k ) F i (z) lim h 0 + h = lim h 0 + F i (0,.., h,.., 0, y,.., y m k ) h 0.

17 uly, 00 : Persistence LJA R Page of Hence C(z) 0. Again, using the positive invariance of X 0, we have: F (x, y) = F (x, y) F (0, y) = 0 d F F (sx, y)ds = ds 0 x F = ( (sx, y)ds)x = A(z)x, where A(z) = 0 x (sx, y)xds = 0 F (sx, y)ds. x Above, we used the notation F Fi x (z) for ( x j (z)) i,j k. Notice that A(E) = C(E). F is C implies that A(z) is continuous. Now a), b) and c) follow from Lemma.0. Notice that, in both Lemma.0 and Corollary., the fact that the components of E that are considered to be zero are chosen to be the first components, is not important, since this can be anyway achieved by reordering the equations. f A(E) is positive and irreducible (hence A(E) T is so), then according to the Perron- Frobenius theory, r(a(e)) = r(a(e) T ) is a positive real eigenvalue of A(E) T, and there exists a strictly positive eigenvector of A(E) T corresponding to r(a(e)). We will use this observation repeatedly in the following proofs (whenever Lemma.0 or Corollary. are applied), without further reference.. Proofs Throughout this section, we will make repeated use of the following notation: M K = p LK 0 0 q LK p JK 0, K =,, R. 0 q JK p AK Also, let P be the diagonal matrix having p L, p J, p A on the main diagonal; D be the diagonal matrix having p L, p J, p A on the main diagonal; and v := (v L, v J, v A ). For a vector x = (L, J,..., F ) R+ 0 (R+) let x K = (L K, J K, A K ) T, K =,, R (K =, ). We mention that in the proofs of the persistence results, we will use the sets 0 (see ()) and Ω (see ()) together with Theorem.. Proof. (of Proposition.) We present here only the proof for the model (), the proof for the model () being completely analogous. We will apply [, Theorem...]. First, we show that there exists a bounded set that attracts all orbits of (). Let x = (L, J,..., F ) T R 0 +, and M = sup xφ(x). Then where x x n+ Ax n + f, n 0, () A = M O O O D M O O O P M R O O b F v O p F

18 uly, 00 : Persistence LJA R Page of and f = (M( b c A + b R c AR ), 0, 0, M b c A, 0, 0, 0, 0, 0, 0) T. From () we get n x n A n x 0 + A i f, n. i=0 n Using that A is an invertible matrix we have that A i = ( A) ( A n ), hence i=0 x n A n x 0 + ( A) ( A n )f, n Clearly, r(a) <, hence A n O as n. o we have that the set [0, w] := {x R 0 + x w}, where w = ( A) f, attracts all points in R 0 +. A simple computation shows that w 0. Next, we show that orbits of bounded sets are bounded. Let B R 0 + be a bounded set and x 0 B. o B, the closure of B, is compact. ince A n O as n, there exists a matrix C O and N N such that A n < C and ( A) ( A n ) < C, n N. Then, from (i) we get x n C(x 0 + f), n N. Let M B := max x, and b := C(M B + f). Then we have that x B x n b, n N, x 0 B. The state space being R 0 +, it is trivial to check that () is asymptotically smooth (see [] for a definition of this). Now, from [, Theorem...], we have that () has a global attractor of bounded sets. Proof. (of Lemma.) The characteristic polynomial for the eigenvalues of A K is (i). p(λ) = (λ p LK )(λ p JK )(λ p AK ) q LK q JK b K. uppose K >. Then p() < 0. But lim inf p(λ) = and since p is continuous, λ there exists ˆλ > such that p(ˆλ) = 0. Hence r(a K ) >. Now suppose K <. Then p() > 0. t is trivial to check that p (λ) > 0, λ >. Hence we have p(λ) > p(), λ > p(λ) > 0, λ (i). But since A K is positive and irreducible, its spectral radius is an eigenvalue. Then from (i) we conclude that r(a K ) <, and the lemma is proved. Lemma.. Assume K >, DQ K (x) > 0, x R +, and xφ(x) bounded on [0, ). Then ( L, J, Ā) is an asymptotically stable fixed point of Q K, and attracts the orbits of all points in R + \ {0}. Proof. Let E Q K := ( L K, J K, ĀK). First we show that E Q K is asymptotically stable. We can write Q K (x) as A K (x)x (A K (x) given in ()). Notice that DQ K (E Q K ) < A K (E Q K ), and since both matrices are positive and irreducible, we obtain that r(dq K (E Q K )) < r(ak (E Q K )) (i). EQ K being a fixed point of Q K implies

19 uly, 00 : Persistence LJA R Page of (E Q K )T = A K (E Q K )(EQ K )T, so is an eigenvalue of A K (E Q K ), and since the corresponding eigenvector E Q is strictly positive, from the Perron-Frobenius theory we have that r(a K (E Q K )) =.Then, using (i) we have that EQ K is asymptotically stable. Now let x 0 > 0. We have that x n+ = Q K (x n ) M K x n + f, n 0, where f = ( b K c AK M, 0, 0) T, with M = sup xφ(x). Then, as in the proof of Proposition., we obtain that the orbit of x 0 is attracted to x [0, w] := {x R + x w}, where w = ( M K ) f 0, () and so ω(x 0 ) [0, w]. ince x 0 > 0 was arbitrarily chosen, we have that E Q K [0, w]. Notice that M K w + f = w, hence Q K (w) w. Let now x ω(x 0 ). As it will be proved in Theorem., 0 ω(x 0 ). o x > 0. ince Q K (x) n 0, n, without a loss of generality we can assume that x 0. We want to show that we can find u 0 such that u x and Q K (u) > u. DQ K (0) = A K, hence λ := r(dq K (0)) > (from Lemma.). DQ K (0) being irreducible, it has an eigenvector ũ 0, corresponding to λ, and let u = sũ, s > 0. Using Taylor expansion and the fact that 0 is a fixed point of Q K we get Q K j (u) = i= Q K j x i (c j )u i j =,, (ii) where c j s are points on the line segments joining 0 and u. Let ε > 0 such that (λ )u j ε u > 0; j =,, (iii). uch an ε exists, because λ >. Q K being a C map, DQ K is continuous, so we can choose s sufficiently small (hence u and c j, j =,, are small) such that to have both Q K j x i and u x. Using (ii) we have (c j ) Q K j (0) ε, i, j =,, x i Q K (u) DQ K (0)u Υu > u λu Υu > u (iii) holds, where we denoted by Υ the matrix having each element equal to ε. o we have u x w and u Q K (u), Q K (w) w. DQ K (x) > 0, for any x R +, implies that Q K is a monotone map (see []). Thus, (Q n K u) n is an increasing sequence, hence it is convergent (see []) and, by continuity of Q K, (Q n K (u)) n converges to a fixed point of Q K. But that fixed point must be E Q K. imilarly, (Q n K (w)) n is a decreasing sequence, and Q n K (w) EQ K, n 0. o (again, see []), it converges to E Q K. Also, we have Qn K (u) Qn K (x) Qn K (w), hence Qn K (x) EQ K as n. Now, ω(x 0 ) being compact and invariant, we have that E Q K ω(x0 ). But then, since E Q K is asymptotically stable, we conclude that ω(x0 ) = E Q K.

20 uly, 00 : Persistence LJA R Page of Remark. n the above lemma, the map Q K needs not be monotone in the whole R +. As it can be noticed in the proof, only monotonicity in [0, w] is used, given that ( L K, J K, ĀK) is in [0, w]. Proof. (of Theorem.) Again, let E Q K := ( L K, J K, ĀK). First we show that any solution is attracted to the set [0, w] = {x R + x w}, where w = φ ( p AK )(, p AK, ) T. () K c AK q LK q JK c AK q JK c AK p AK c AK q LK q JK, pak c AK q JK, Denote ( c AK ) T by c. Let y 0 > 0, and x 0 = y 0. Let M = sup xφ(x), and a = ( bk c x AK M, 0, 0) T. Then, as in the proof of Proposition., we obtain that ω(y 0 ) [0, w ], where: w = ( M K ) a = M K c. Now let y ω(y 0 ), and again, consider the solution to (0) starting at x 0 = y. Then, since ω(y 0 ) is invariant, x n [0, w ], and so we have x n+ M K x n + a, where a = (M b K c AK, 0, 0) T, with M = ω(y ) [0, w ], where max x [0,M K] w = ( M K ) a = M K c. xφ(x). imilarly, we get that Note that ω(y ) ω(y 0 ) [0, w ], so ω(y 0 ) [0, w ]. Continuing inductively, we obtain that ω(y 0 ) [0, w n ], n 0 (i), where w n = M n K c, with M n = max xφ(x). t is clear that, for both choices of φ, M M. x [0,M n K] uppose M i M i. Then M i+ = max xφ(x) max xφ(x) = M i. x [0,M i K ] x [0,M i K ] Thus, by induction, we have M n+ M n, n, which implies that (M n ) n is a convergent sequence. Hence (w n ) n is convergent, and let w be its limit. Notice that lim > 0, because if it is zero, then w = 0, which implies n M n ω(y 0 ) = {0}, contradiction to part b) of Theorem.. o, we have w 0. n the Beverton-Holt case, x xφ(x) = x/( + x) is increasing on R +, while in the Ricker case, x xφ(x) = xe x is increasing on [0, ], and in this latter case we have M n K M K = K /e, n. o, in any case, M n = max xφ(x) = M n K φ(m n K ) = c AK (w ) n φ(c AK (w ) n ). x [0,M n K] Thus, Letting n go to infinity we get w n = K c AK (w ) n φ(c AK (w ) n )c. w = K c AK w φ(c AK w )c.

21 uly, 00 : Persistence LJA R Page 0 of olving the above equation, and taking into account that w 0, we obtain w = φ ( K )c. From (i) it follows that ω(y 0 ) [0, w]. ince y 0 > 0 was arbitrarily chosen, we have that E Q K [0, w]. From Theorem. part b) we have that 0 ω(y 0). We now discuss a) and b) simultaneously. DQ K (x) = a (x) a (x) a (x) q LK p JK 0 0 q JK p AK where a (x) = p LK + c LK b K φ (T )x, a (x) = c JK b K φ (T )x = 0 (because c JK = 0), and a (x) = b K [c AK φ (T )x + φ(t )]. We want to show that the system is monotone in [0, w]. As we mentioned before, a sufficient condition for this is DQ K (x) > 0, x [0, w]. o let x [0, w]. n the Beverton-Holt case, i.e. φ(x) = /( + x), we have, a (x) = p LK c LK b K x ( + c LK x + c JK x + c AK x ) = p LK c LKb K c AK c AK x ( + c LK x + c JK x + c AK x ) p LK c LKb K c AK c AK x ( + c AK x ). But x [0, w] means that c AK x φ (/ K ) = ( K ). The function x x/( + x) is increasing on [0, ] and decreasing on [, ). Thus, if K, the maximum of c AK x /( + c AK x ) is ( K )/ K, otherwise it is /. n either case, from our hypotheses, we have a (x) 0. t is trivial to check that a (x) 0. n the Ricker case, i.e. φ(x) = e x, notice that c AK x φ (/ K ) = ln( K ), x [0, w]. o the maximum of e c AKx c AK x is ln( K) K, because x xe x is increasing on [0, ]. Thus, we have a (x) = p LK c LK b K e T x p LK c LK c AK b K e cakx c AK x p LK c LKb K ln( K ) 0 c LKb K c AK K c AK p LK K ln( K ). Also, a (x) = q LK b K e T ( c AK x ) 0, because c AK x φ (/ K ) = ln( K ) ln e = 0. o, the system is monotone in [0, w], and sice E Q K [0, w], the conclusion follows from Lemma. (see Remark above).. Proofs of results for LJA- model Proof. (of Proposition.) Let E = (L, J,..., F ) be a boundary fixed point of (). Notice that E (B F F -axis) L, J, A, F > 0, and E B L, J, A > 0. Hence E B F B. a) The dynamics restricted to B F are given by (0), and a simple computation shows that ( L, J, Ā) is the unique fixed point of this reduced model (see also [], page ).

22 uly, 00 : Persistence LJA R Page of b) The dynamics restricted to B are given by L n+ = p L L n + b φ(t n )A n J n+ = q L L n + p JJ n A n+ = q J J n + p AA n () F n+ = b F (v L L n + v JJ n + v AA n ) + p F F n The first three equations are decoupled from the fourth, and form a system completely analogous to (0) (with replaced by ). o, from a), ( L, J, Ā) is the only non-zero fixed point of it. But then solving F = b F (v L L +v J J +v A Ā )+p F F for F we get F = F. Proof. (of Theorem.) a) ee [] b) Let ρ(x) = min{l, J, A }. The following holds: x 0 > 0 x n 0, n o 0 = {0}, and Ω = {0}. Equation (0) can be written in the form x n+ = A (x n )x n. Now we apply Lemma.0, with X 0 = {0} and E = 0. We have that A (0) = A. Thus, r(a (0)) >, because > (see Lemma.). Part a) of Lemma.0 implies that {0} is isolated, and because R +\X 0 is positively invariant, from part c) we have that W (0) = {0}. Acyclicity of {0} is trivial in this case. Hence, from Theorem. we obtain that (0) is uniformly ρ-persistent. Using (i), this is equivalent to: there exists ε > 0 such that lim inf n ρ(xn ) > ɛ, x 0 R + \ {0}. Proof. (of Remark, ection.) a) The proof for this part is trivial. b) First assume ( L, J, Ā) is an asymptotically stable fixed point of Q and attracts the orbits (under Q ) of all points in R + \ {0}. The system () restricted to B is given by (). Let x 0 B \ F axis. Then x 0 > 0 and xn+ = Q (x n ), n 0. o xn ( L, J, Ā) T. Let ε > 0. There exists N N such that L n L < ε( pf ) b F v L, J n J < ε( pf ) b F v J and A n Ā < ε( pf ), n N. Then, (i). b F v A ε( p F ) + b F v L L n ε( p F ) + L b F v L, n N, L ε( p F ) + b F v J J n ε( p F ) + J b F v J, n N, and J ε( p F ) b F v A + Ā A n ε( p F ) b F v A + Ā, n N. From the inequalities above and the equation for F we obtain ε( p F ) + ( p F ) F F n+ p F F n ε( p F ) + ( p F ) F, n N.

23 uly, 00 : Persistence LJA R Page of By iterating this from n = N to n = N + k, we get p k F (F N F + ε) ε F N+k F p k F (F N F ε) + ε, k 0 which implies ε lim inf k F N+k F ε, and ε lim sup F N+k F ε. k ince ε was arbitrarily chosen, we conclude that F n F as n. Now we will show that E is stable in B. Let ε > 0. We can choose ε as above, such that ε < ε. ince ( L, J, Ā) is asymptotically stable for Q, there exists 0 < δ < ε such that (L 0, J 0, A0 ) ( L, J, Ā) < δ (L n, J n, An ) ( L, J, Ā) < min{ε, ε( pf ) b F v L, ε( pf ) b F v J, ε( pf ) b F v A }, n 0. Let now x 0 B such that x 0 E < δ. Then (L 0, J 0, A0 ) ( L, J, Ā) < δ and F 0 F < δ. o, as above, (i) holds with N = 0, and we get F n F ε, n 0. Hence x n E = (L n, J n, An ) ( L, J, Ā) + F n F ε < ε, n 0, hence E is stable in B. The other implication is trivial. Proof. (of Theorem.) a) We have that x n+ A x n, n 0. But < r(a ) < (see Lemma.). Hence x n 0. b) As mentioned in the beginning of ection, the spectral radius of the Jacobian J(0) of () evaluated at 0 is max{r(a ), r(a ), p F }. o using Lemma., we have that 0 is asymptotically stable if < and <, and unstable if > or >. The following inequality holds: x n+ A O O D A O x n. O b F v p F Using Lemma., we see that the spectral radius of the matrix above is less than one, hence x n 0. c) The Jacobian of () evaluated at E is of the form J(E ) = (i) ( ) C O J F. () o, its set of eigenvalues consists of eigenvalues of C and J F. ince E is asymptotically stable in B F, r(c). f r(c) < then clearly E is asymptotically stable in R +, because r(j F ) <. f r(c) =, then () has a center manifold included in the positively invariant set B F. But again, since E is asymptotically stable in B F, we conclude that E is asymptotically stable in R + (see [, Theorem..]).

24 uly, 00 : Persistence LJA R Page of Now assume r(ĵ F ) < and Q as in Theorem.. Using that e x x, x 0, notice that x n+ A(x n )x n, n 0 (i), where A(x n ) is obtained from J F by replacing φ( T ) by and L, J, Ā respectively by L n, J n, An. The following inequality holds: x n+ M x n + ( b c A M, 0, 0) T, where M = sup xφ(x). o, as in the proof of Theorem., we get that x n is x attracted to the box [0, w] (ii), where w is given in (). From the above mentioned theorem we know that Q is monotone in [0, w], Q (w) w and Q n (w) ( L, J, Ā) T as n. Let now x 0 R + \ B. uppose that ω(x 0 ) B F ( ). o let y 0 ω(x 0 ) \ B F. ince ω(x 0 ) [0, w] (see (ii)), we have y 0 w. f y0 = 0 then there exists N 0 such that x N w x N+ Q (x N ) Q (w) w. By induction we get that x n w, n N. f y 0 > 0 then y 0, and because y B F, we have y Q (y ) w, which implies that there exists N 0 such that x N < w xn w, n N. Thus, in any case, there exists N 0 such that x n w, n N. Then, x N+k Q k (w), k 0. But Qk (w) ( L, J, Ā) T as k. Hence, ε > 0 N > 0 such that x n ( L + ε, J + ε, Ā + ε) T, n N. This implies that A(x n ) A ε := A( L + ε, J + ε, Ā + ε), n N, and from (i) we get A ε x n, n N, from which we obtain: x n+ x N+k A k εx N, k 0 (iii). Notice that A 0 = Ĵ F. Because A(x) is a continuous matrix, and r(ĵ F ) <, we can choose ε so small such that r(a ε ) <, and then from (iii) we conclude that x n 0, x 0 R + \ B (iv). This implies that ω(x 0 ) B F, hence we have a contradiction to ( ). o, in fact we have that ω(x 0 ) B F. Assuming, from Theorem. part (), we have that 0 ω(x 0 ). Then (iv) implies that ω(x 0 ) contains a point in B F \ {0}, which implies, using (C ) and the fact that ω(x 0 ) is closed and invariant, that E ω(x 0 ). ince E is asymptotically stable in R + (see Theorem.), we conclude that ω(x 0 ) = E. d) The Jacobian of () evaluated at E is of the form ( ) J J(E ) = O. () D The proof for asymptotic stability of E is completely analogous to the proof of asymptotic stability of E in part c). Now suppose that (C ) holds and <. Let x 0 R +\(B F F axis). From part a) we have that ω(x 0 ) B. Theorem. () implies that ω(x 0 ) F axis =. Hence ω(x 0 ) B \F axis. ince any solution starting in B \F axis converges to E (by (C )), and ω(x 0 ) is a closed and invariant set, we have that E ω(x 0 ). But because E is asymptotically stable, we conclude that ω(x 0 ) = {E }. Proof. (of Theorem.) () a) Let ρ(x) = L + J + A + L + J + A. Then

25 uly, 00 : Persistence LJA R Page of = {x 0 R + ρ(x n ) = 0, n 0} = F axis, which is a closed and positively invariant set. Ω = x 0 ω(x) = {0} (i). We want to apply Lemma.0 with X 0 = {x R + L = J = A = 0} and E = 0. The following inequality holds: x n+ = à (x n )x n, () with à (x n ) defined on page. o à (0) = A, and by Lemma. we have that r(a ) >. Thus, from part a) of Lemma.0, we have that there exists a neighborhood U of 0 in which there is no positively invariant set containing a point x with L > 0, or J > 0, or A > 0. Now we are applying the same lemma, this time with X 0 = {x R + L = J = A = 0} and E = 0. We have x n+ A (x n )x n. () Now A (0) = A, and so r(a (0)) > (again, from Lemma.). Hence there exists V U a neighborhood U of 0 in which there is no positively invariant set containing a point x with L > 0, or J > 0, or A > 0. n conclusion, there is no positively invariant set in V containing points in R + \ F axis. But now, 0 being asymptotically stable on the F axis, we obtain, from (the proof of) Lemma.0 part b), that 0 is isolated in R + (ii). ince R +\F axis is positively invariant, from the (proof of) part c) of the same lemma we have that W (0) F axis (iii). Acyclicity of {0} comes from Lemma. (because, as it can be easily checked, 0 is asymptotically stable on the F axis), and using it together with (i), (ii) and (iii) we get, by Theorem., that () is uniformly ρ-persistent, i.e. ε > 0 s.t. lim inf n Ln + L n + J n + J n + A n + A n > ε, x 0 R + \ F axis. b) We first show that () is uniformly ρ-persistent, where ρ(x) = min{l, J, A }. t is trivial to check that for any x 0 satisfying x 0 > 0, we have x n 0, n (i). o 0 = {x 0 R + ρ(x n ) = 0, n 0} = B, which is a closed and positively invariant set. A solution to () restricted to B satisfies x n+ A x n. () But < r(a ) < x n 0 F n 0. o, we conclude that Ω = x 0 ω(x) = {0} (ii). Now we want to apply Lemma.0 with X 0 = {x R + L = J = A = 0} = B and E = 0. Equation () applies here, and using that à (0) = A, we have r(ã (0)) > (see Lemma.). The Jacobian at 0 of () restricted to X 0 is ( A 0 b F v p F o, its set of eigenvalues consists of eigenvalues of A, and p F, all of which are strictly less than. Hence 0 is asymptotically stable in X 0 = 0. This implies (see Lemma.) that {0} is acyclic (iii), and that (see Lemma.0 b)) {0} is isolated in R + (iv). ince R + \ X 0 is positively invariant, we also have that W (0) X 0 = B = 0 (v). From (ii)-(v) we get, using theorem., that () is uniformly ρ-persistent. Using (i), this is equivalent to: ). ε > 0 s.t. lim inf n min{ln, J n, A n } > ε > 0, x 0 R + \ B.