# Discrete Models of Disease and Competition

Authors: R. Bravo de la Parra; M. Marvá; E. Sánchez; L. Sanz
Journal: Discrete Dynamics in Nature and Society
 (2017)

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## Abstract

The aim of this work is to analyze the influence of the fast development of a disease on competition dynamics. To this end we present two discrete time ecoepidemic models. The first one corresponds to the case of one parasite affecting demography and intraspecific competition in a single host, whereas the second one contemplates the more complex case of competition between two different species, one of which is infected by the parasite. We carry out a complete mathematical analysis of the asymptotic behavior of the solutions of the corresponding systems of difference equations and derive interesting ecological information about the influence of a disease in competition dynamics. This includes an assessment of the impact of the disease on the equilibrium population of both species as well as some counterintuitive behaviors in which although we would expect the outbreak of the disease to negatively affect the infected species, the contrary happens.


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## Body

## 1. Introduction

Species interactions are a central issue in ecology in order to explain community structure and its dynamics. Among the frameworks that try to explain ecosystems dynamics, ecoepidemiology has become a proper discipline in its own right [1, 2]. Loosely speaking, its main objective is to understand the effect of diseases or parasites on species interactions and vice versa. The empirical evidence of such an influence has motivated a great deal of mathematical modeling in order to reveal the underlying mechanisms.In this work we focus on ecoepidemic competition models that have attracted great attention [3]. Empirical observations have revealed that disease/parasites can affect the outcome of species competition in several different ways. For instance, a shared parasite between the competing species can change the fate of competition, allowing the otherwise inferior competitor to rule out the superior competitor, as observed by Park [4] withTribolium castaneum (the stronger competitor),Tribolium confusum, and the parasitoidAdelina tribolii that lowers the fecundity ofT. castaneum. There is also evidence of species coexistence induced by specialist (i.e., affecting only one competitor) parasite. This is the case ofAnolis lizards in the Caribbean area [5].Anolis gingivinus is a stronger competitor thanA. wattsi and coexisting populations are found only in the presence of the parasitePlasmodium azurophilum. This parasite that reduces the competitive abilities of the host rarely affectsA. wattsi but can be common inA. gingivinus.It is usually assumed that disease/parasites reduce either the growth or competitive abilities of infected individuals or both of them. Nevertheless, this is not always the case. Regarding their effect on growth, some parasites spread through host offspring, so that the strategy consists on enhancing host fecundity to improve their spreading. For instance, this is the case ofWolbachia and its hostBemisia tabaci [6, 7].Bemisia t. competes withTrialeurodes vaporariorum (Westwood) [8] and it has been reported that there are no evidences of Westwood infected byWolbachia [9]. As far as competitive abilities are concerned, a laboratory study [10] dealt with the effect of the tapewormHymenolepis diminuta on competition betweenT. castaneum andT. confusum.T. castaneum (as we have said, the superior competitor) has higher infection and fitness costs from parasitism, so that a parasite-mediated reversal of the outcome of competition was expected. Nevertheless,T. confusum was excluded even faster than what had happened in the absence of infection; it was suggested that intraguild predation is behind this outcome [10].From the mathematical point of view, complete ecoepidemic competition models are known for the difficulty of their analytical study (see [3] box 2.3 and references therein). There exist a number of studies in continuous time [3] but, to our knowledge, our work is the first attempt in discrete time.An ecoepidemic competition model combines three different processes: each species demography, intra- and/or interspecies competition, and epidemics. To our knowledge, all the ecoepidemic competition models in the literature consider that the three processes evolve within the same time scale. In other words, it is assumed that the effects of demography, competition, and infection are accounted for at the population level. This assumption is fairly true for certain long-term infections (as AIDS, tuberculosis, etc.) but is not for some others. Indeed, there are diseases such that a number of infection/recovery episodes take place within each demography period. This assumption is equivalent to that of the existence of different time scales associated with each process. An example of this scenario is thehuman trachoma, an infectious disease that is the leading cause of blindness worldwide and is caused by a microorganism calledChlamydia trachomatis. The immune system of human beings can recover but reinfections happen often [11]. After repeated episodes, successive scars cause the edge of the eyelid to fold inward, causing pain and even permanent damage to the cornea which may lead to irreversible blindness. In fact, the assumption of an endemic background scenario can be understood as the effect on the population dynamics of the disease having attained one of its equilibrium states.In this work we consider two models of competition and disease. The first one is a single species model in which there is intraspecific competition and the population is affected by a disease-parasite. Demography-competition is assumed to be governed by an adaptation of the Beverton-Holt model and the disease corresponds to a discrete SIS epidemic model with frequency-dependent transmission [12]. The second model corresponds to two competing species, one of them affected by specialist disease/parasite. Competition is built up from the classical discrete time competition Leslie-Gower model [13], and the disease is again introduced by means of a discrete SIS epidemic model. This ecoepidemic model happens to be rich enough to exhibit the complex behaviors described above but simple enough to be analytically tractable.One of the differences between continuous and discrete models is that in the former all processes involved in the model (demography, competition, and infection/recovery) occur instantaneously at the same time, whereas in the later it is usual to consider that processes take place sequentially [14]. Keeping this idea as well as the separation of time scales in mind, in the construction of both models we assume that a large number k of epidemic-related events take place within each demographic/competition step. Concentrating our attention on the case of the two competing populations, the resulting model is a three-dimensional system. Making use of the existence of different time scales, it is possible to build up a reduced two-dimensional system that approximates the behavior of the original system and that simplifies the analysis of the model (see [15] for a recent review in the field of discrete approximate reduction techniques). The resulting reduced system can be regarded as a generalization of the Leslie-Gower competition model and retains some of its properties, as the fact that the unique attractor states are equilibrium points. However, the reduced model allows for disease-induced multistability scenarios.This work is organized as follows: in Section2 we consider the interplay between intraspecific competition and a disease on a single species model. In Section 3 we introduce a competing species and carry on with the mathematical analysis of the resulting system and its ecological interpretation. Subsequently, we analyze separately the scenarios of a disease affecting either species growth or species competitive abilities. The conclusions in Section 4 and Appendix with the proof of the mathematical results complete the manuscript.


## 2. Disease and Intraspecific Competition

We consider a host population with density dependent regulation that is affected by a disease which acts on a shorter time scale than the demographic dynamics. The time unit of the discrete model is the one associated with its demographic part. We consider that, in this time unit, called slow, a single episode of demographic change following a numberk of disease infection-recovery cycles happens. In this way we intend to describe the fact that pathogens exhibit outbreaks on short time scales, on the order of days or weeks, whereas demographic changes might be considered annual.Time in the slow time unit is denoted byt. NS(t) and NI(t) represent the susceptible and the infected individuals, respectively, at time t. Between time t and time t+1 we consider the disease dynamics acting sequentially k times followed by one demographic episode.The disease dynamics is defined by means of the discrete time SIS epidemic model studied in [12] whose associated map is (1)FNS,NI=FSNS,NI,FINS,NI=NS-βNSNINS+NI+γNI,NI+βNSNINS+NI-γNI,so that the total effect of disease during one slow time unit is represented by its kth iterate that we denote as F(k).We assume that transmission is frequency-dependent, withβ being the transmission coefficient. The recovery coefficient γ represents the fraction of infected individuals that recover in a unit of time.The disease dynamics keeps constant the population size(2)FSNS,NI+FINS,NI=NS+NI.Henceforth we assume that the following inequalities hold:(3)β<1+γ2,γ≤1,so that if we denote R+2=0,∞×0,∞ and R∘+2=(0,∞)×(0,∞), one has F(R+2)⊂R+2 [12]. Note that condition γ≤1 amounts to requiring that the number of recovered individuals in one time step can not be greater than the number of infected individuals, which clearly is necessary for the number of infected individuals to remain positive. The second condition in (3) says that the transmission coefficient can not be too high with respect to the recovery coefficient because, if this were not so, the number of susceptibles might become negative.The asymptotic behavior of the solutions of the discrete system defined by mapF (1) is studied in [12]. The basic reproduction number for this model is R0=β/γ. If R0≤1 then for any positive initial condition the number of infected individuals monotonically decreases to 0; that is, positive solutions converge to the disease-free equilibrium N0∗=(N,0), where N is the constant population size. On the other hand, if R0>1 the disease becomes endemic. Assuming, in addition to inequalities (3), that (4)γ<β≤2+γ,positive solutions converge to an asymptotically stable endemic equilibrium (5)Ne∗=NS∗,NI∗=γβN,1-γβN.If2+γ<β<1+γ2 then the monotonic convergence to an endemic equilibrium disappears giving rise to period-doubling and chaotic behavior.In the following we will restrict to the case where (4) holds and therefore the endemic equilibrium is stable. The fact that the fast disease process attains an equilibrium point in the long term allows one to reduce the dimension of the system that we are proposing. This equilibrium is expressed as the limit, F-, of the iterates of map F(6)F-NS,NI≔limk→∞FkNS,NI=νN,1-νN,where N≔NS+NI and ν is defined by (7)ν≔1,ifβ≤γ,γβ,ifγ<β≤2+γ.Note that ν=1 implies that the disease is eradicated.To build up our demographic model where we have to take into account susceptible and infected individuals, we start from the Beverton-Holt model [16] (8)Nt+1=bNt1+cNt,where the positive parameters b and c are the intrinsic growth rate and the intraspecific competition coefficient. There are two possible asymptotic behaviors of the solutions of equation (8) with positive initial condition:(1)
Ifb≤1 then limt→∞N(t)=0.(2)
Ifb>1 then limt→∞N(t)=b-1/c.Note that this corresponds to a logistic behavior with carrying capacity(b-1)/c.Now we adapt the Beverton-Holt model to consider individuals classified into susceptible and infected. The trait-mediated indirect effects of parasites on hosts are considered both in the growth and in the intraspecific competition. We assume different intrinsic growth rates for susceptible,bS, and infected, bI, individuals. We also distinguish four intraspecific competition coefficients: cSS, cSI, cIS, and cII. The resulting model is (9)NSt+1=bSNSt1+cSSNSt+cSINIt,NIt+1=bINIt1+cISNSt+cIINIt.To build up the complete model combining the demographic and the disease processes we compose thekth iterate F(k) of map F (1) and the map defining system (9). Using the notation F(k)(NS,NI)=FS(k)(NS,NI),FI(k)(NS,NI) the complete model has the form (10)NSt+1=bSFSkNSt,NIt1+cSSFSkNSt,NIt+cSIFIkNSt,NIt,NIt+1=bIFIkNSt,NIt1+cISFSkNSt,NIt+cIIFIkNSt,NIt.System (10) can be reduced to a scalar equation from which we can approximate the asymptotic behavior of its solutions. To do so we make use of discrete approximate reduction techniques and follow the reduction procedure presented in [15, 17]. Assuming that the disease process has attained its equilibrium (6), the dynamics of the total number of individuals in the population, N=NS+NI, can be approximated by the following equation:(11)Nt+1=bSνNt1+cSSνNt+cSI1-νNt+bI1-νNt1+cISνNt+cII1-νNt,obtained by adding up the two equations in (10) and using the following approximations: NS≈νN and NI≈(1-ν)N [18].We can write (11) in the next simplified form (12)Nt+1=b1Nt1+c1Nt+b2Nt1+c2Nt,with four parameters defined as b1=bSν, b2=bI(1-ν), c1=cSSν+cSI(1-ν), and c2=cISν+cII(1-ν). The asymptotic behavior of its solutions is stated in the next theorem.Theorem 1.
One considers equation (12) with b1, c1, and c2 being positive and b2 nonnegative. Let N(t) be any solution of (12) with positive initial condition:(1)
Ifb1+b2≤1 then limt→∞N(t)=0.(2)
Ifb1+b2>1 then limt→∞N(t)=N-, where (13)N-=12c1c2b2-1c1+b1-1c2+1-b2c1+1-b1c22+4b1+b2-1c1c2.Proof.
See Th. A.4 in [18].Making use of the last theorem and the results on approximate aggregation in [18], we can decide the asymptotic behavior of solutions of system (10). We assume that the disease becomes endemic (R0>1, i.e., ν<1) because otherwise system (10) behaves as the scalar Beverton-Holt model. Therefore we have that, for any positive initial condition, the solution (NS(t),NI(t)) of system (10) verifies the following:(1)
IfbSν+bI(1-ν)≤1 then it approximately tends to (0,0).(2)
IfbSν+bI(1-ν)>1 then it approximately tends to WN-/R0,(1-1/R0)N- where W is the transformation associated with the competition process (9); that is, (14)WN-R0,1-1R0N-=bS1N-R01+c1N-,bI11-1/R0N-1+c2N-.Note that if the disease is established, either the population tends to extinction or it tends to an endemic equilibrium. To analyze the influence of the parasite on the population dynamics we compare the asymptotic behaviors of the solutions of (8), representing the dynamics of the disease-free population, and (11), which yields the approximate long-term behavior of system (10). For the comparison we assume that parameters affecting susceptible individuals in system (10) coincide with the corresponding ones in (8), that is, b=bS and c=cSS.It is straightforward to check thatmax⁡{bS,bI}<1 implies extinction and min⁡{bS,bI}>1 endemic stabilization. On the other hand, if bS>1, that is, if the population without disease tends to (bS-1)/cSS, then the disease leads the population to extinction for bI small enough, specifically when bI<(1-bSν)/(1-ν). Conversely, if the population is not viable without disease, bS<1, a large enough infected growth rate, bI>(1-bSν)/(1-ν), entails the population attaining equilibrium N-/R0,(1-1/R0)N-.If the population attains a positive equilibrium in both cases, with and without disease, we can explore the influence of the disease by comparing the two final population sizes. Without disease it is(bS-1)/cSS and with disease N-. We do the comparison in the relevant particular case in which the disease does not affect the competitive abilities of individuals, that is, cSS=cSI=cIS=cII=c. In this case we obtain the ratio (15)N-bS-1/c=bSν+bI1-ν-1bS-1.As a function of the growth rate of infected individuals, bI, this ratio is increasing and equals 1 for bI=bS. More interesting is its dependence on ν=1/R0(16)N-bS-1/c=bS-bIbS-1ν+bI-1bS-1.If bS>bI then it increases with ν; that is, the larger R0, the smaller the ratio. Conversely, if bS<bI the larger R0, the larger the ratio.


## 3. Disease and Interspecific Competition

### 3.1. Model Construction and Reduction

In this section we generalize the setting of the previous one and introduce a second species, which we consider disease free, which competes, according to the well-known Leslie-Gower model [13], with the species which is affected by the disease. Specifically, let us consider two species that we denote by 1 and 2, in competition and a disease defined by system (1) and which affects only species 1. Let NS1 and NI1 represent the number of susceptible and infected individuals of species 1 and let N2 be the amount of individuals of species 2. Then, the slow dynamics is described by(17)NS1t+1=bS1NS1t1+cSSNS1t+cSINI1t+cS2N2t,NI1t+1=bI1NI1t1+cISNS1t+cIINI1t+cI2N2t,N2t+1=b2N2t1+c2SNS1t+c2INI1t+c22N2t,where all growth and competition coefficients are assumed to be positive.As in Section2, we consider that in the time unit of the model a single episode of demographic change happens, defined by (17), followed by a number k of disease infection-recovery cycles corresponding to the SIS model (1) introduced in Section 2. Then, the complete system that takes into account the joint effect of the slow and fast processes takes the form (18)NS1t+1=bS1FSkNS1t,NI1t1+cSSFSkNS1t,NI1t+cSIFIkNS1t,NI1t+cS2N2t,NI1t+1=bI1FIkNS1t,NI1t1+cISFSkNS1t,NI1t+cIIFIkNS1t,NI1t+cI2N2t,N2t+1=b2N2t1+c2SFSkNS1t,NI1t+c2IFIkNS1t,NI1t+c22N2t.If we follow the reduction procedure presented in [15, 17, 18], which essentially consists in adding up the first two equations in (18) and using the approximations NS1≈νNS1+NI1 and NI1≈(1-ν)NS1+NI1, we obtain the reduced system(19)N1t+1=bS1νN1t1+cS1N1t+cS2N2t+bI11-νN1t1+cI1N1t+cI2N2t,N2t+1=b2N2t1+c21N1t+c22N2t,where N1≔NS1+NI1, cS1≔cSSν+cSI1-ν, cI1≔cISν+cII1-ν, and c21≔c2Sν+c2I1-ν.Note that system (19) generalizes the classical Leslie-Gower competition model [13] in the sense that at ν=1 (i.e., the disease-free scenario) system (19) becomes the usual Leslie-Gower competition model.


### 3.2. Model Analysis

In order to carry out the mathematical treatment of system (19) we express it in the form (20)N1t+1,N2t+1=HN1t,N2t,where H is the map defined by(21)Hx1,x2=H1x1,x2,H2x1,x2=ϕ1x1,x2x1,ϕ2x1,x2x2,ϕ1x1,x2≔rS1+cS1x1+cS2x2+rI1+cI1x1+cI2x2,ϕ2x1,x2≔r21+c21x1+c22x2,rS≔bS1ν,rI≔bI11-ν,r2=b2.Note that all the parameters of the model are positive except possibly rI that is nonnegative. As we will see, the numbers (22)ϕ10,0=rS+rI=bS1ν+bI11-ν,ϕ20,0≔r2=b2will play an important role in the dynamics of the system.We note that the functional form of system (20) is a particular case of a model obtained in [19] which studies the dynamics of two spatially distributed populations in competition in which there is fast migration of individuals between the different patches. In that work an analytical study of the system was carried out in the case in which ϕi(0,0)≤1 for at least one value of i=1,2, which essentially corresponds to our Proposition 3 and Theorem 4. In Theorem 6 we provide a full analytical study of the behavior of the orbits of the system in the case ϕi(0,0)>1 for i=1,2, which was only addressed numerically in [19].LetA1=(x1,0):x1>0 and A2=(0,x2):x2>0 be the positive axes. It is immediate to realize that the sets R+2, R∘+2, and Ai, i=1,2 are forward invariant by H. In the sequel, unless otherwise stated we will always assume that we are working on R+2.Let us study the isoclines of the system, that is, the sets defined byx1=H1(x1,x2) and x2=H2(x1,x2). Clearly, besides the positive axes A1 (resp., A2) in which the variable x2 (resp., x1) is constant, the isoclines are the curves Γi defined by ϕi(x1,x2)=1, for i=1,2.Lemma 2.
The isoclinesΓ1 and Γ2 are, respectively, a hyperbola and a straight line. For each i=1,2, Γi intersects R∘+2 if and only if ϕi(0,0)>1 and in that case Γi intersects both the positive axes A1 and A2. Moreover Γ1∩R+2 can be written in the form x2=Φ(x1), x1∈[0,R] for a certain R>0, where Φ is a strictly decreasing convex function.
Except in the degenerate case in whichΓ1 and Γ2 are in fact the same straight line, there can exist at most two positive equilibriums.Proof.
See Appendix.Let us now consider the existence of equilibriums for system (20). Note that E0∗≔(0,0) is an equilibrium point for all values of the parameters. Using Lemma 2 we conclude the follwoing:(i)
For eachi=1,2, there exists a semitrivial equilibrium point Ei∗ on the positive axis Ai if and only if ϕi(0,0)>1. In that case the semitrivial equilibrium is unique.(ii)
A necessary condition for the existence of a positive equilibrium is thatϕi(0,0)>1 for both i=1,2, and in that case there can exist at most two such equilibriums.In what follows we will writex1,x2≤x1′,x2′ (resp., x1,x2<x1′,x2′) to denote that x1≤x1′ and x2≤x2′ (resp., x1<x1′ and x2<x2′). Similarly, we will write x1,x2≤Kx1′,x2′ (resp., x1,x2<Kx1′,x2′) to denote that x1≤x1′ and x2≥x2′ (resp., x1<x1′ and x2>x2′).Proposition 3.
Let us consider system (20).(a)
All solutions inR+2 are forward bounded. More specifically, (23)HR+2⊂S≔0,rScS1+rIcI1×0,r2c22.(b)
H is strongly competitive in R+2; that is, if x,x′∈R+2 are distinct points with x≤Kx′ it follows that H(x)<KH(x′) [20].(c)
All orbits inR+2 are eventually componentwise monotone; that is, for each N1(0),N2(0)∈R+2, the corresponding solution N1(t),N2(t) verifies that Ni(t) is eventually monotone for each i=1,2. Moreover, all orbits tend to an equilibrium as t→∞.Proof.
See Appendix.The next result analyzes the behavior of solutions of system (20) except in the case in which ϕi(0,0)>1, i=1,2.Theorem 4.
Let us consider system (20).(a)
For eachi=1,2, if ϕi(0,0)≤1 then for any initial value on R+2 species i tends to extinction as t→∞.(b)
Ifϕi(0,0)≤1 for i=1,2, all orbits in R+2 tend to E0∗, and if the inequalities are strict then E0∗ is hyperbolic. If ϕi(0,0)>1 for i=1,2, then E0∗ is a repeller and therefore no orbit can converge to E0∗.(c)
Ifϕ1(0,0)>1 and ϕ2(0,0)≤1 then(1) all orbits with N1(0)=0 tend to E0∗ and(2) all orbits with N1(0)>0 tend to E1∗.(d)
Ifϕ2(0,0)>1 and ϕ1(0,0)≤1 then(1) all orbits with N2(0)=0 tend to E0∗ and(2) all orbits with N2(0)>0 tend to E2∗.Proof.
See Appendix.Let us now consider the case in whichϕi(0,0)>1 for i=1,2. In the first place we study the local stability of the semitrivial equilibriums and the behavior of orbits on the positive axes.Proposition 5.
Let us consider system (20) and let us assume that ϕi(0,0)>1 for i=1,2.(a)
All orbits starting on the positivei-axis Ai converge to Ei∗, i=1,2.(b)
Leti=1,2 be fixed. If ϕj(Ei∗)<1, j≠i, then Ei∗ is hyperbolic and attracting. If ϕj(Ei∗)>1, j≠i, then Ei∗ is hyperbolic and unstable.Proof.
(a) Leti=1,2 be fixed. We know that Ai is invariant for H and then any orbit corresponding to an initial condition on Ai must converge to either E0∗ or Ei∗. From Theorem 4 part (b) E0∗ is a repeller and therefore the result follows.
(b) It follows from the usual analysis of the eigenvalues of the corresponding Jacobian matrix. Standard calculations lead to the results bearing in mind thatϕi(Ei∗)=1.In order to study the behavior of solutions for positive initial conditions, we will consider different cases based on the relative position of the intercepts of the isoclines. To this end, we defineRij as the j-intercept of Γi, i,j=1,2. It is immediate to check that Rij is given by(24)R1j=12cSjcIjαj+αj2+4cSjcIjrS+rI-1,j=1,2,R2j=r2-1c2j,j=1,2,where αj≔rS(cIj-1)+rI(cSj-1), j=1,2.Now we distinguish the following scenarios:(i)
CaseA: R11<R21 and R12>R22.(ii)
CaseB: R11>R21 and R12<R22.(iii)
CaseC1: R11>R21 and R12>R22.(iv)
CaseC2: R11<R21 and R12<R22.Taking into account Lemma2, the isoclines divide R∘+2 in a finite number of open connected and disjoint sets Uk, k=1,…,q, in which xi is either strictly increasing or strictly decreasing for each i=1,2. To describe the kind of monotonicity in each region we will use arrows, in such a way that, for example, the situation in which x1 decreases and x2 increases corresponding to a horizontal arrow pointing to the left and a vertical arrow pointing up. Figure 1 shows the different configurations. Note that in case C1 we can distinguish two generic cases depending on whether the two isoclines do not intersect (case C1a) or intersect at two points (case C1b).Figure 1
Different configurations of isoclines and equilibrium points of system (20) when ϕi(0,0)>1 for i=1,2, in terms of the relative position of the intercepts of isoclines, Rij (24).
(a)
Case A. R11<R21 and R12>R22
(b)
Case B. R11>R21 and R12<R22
(c)
Case
C
1
a. R11>R21 and R12>R22 and no positive equilibrium
(d)
Case
C
1
b. R11>R21 and R12>R22 and two positive equilibriums
(e)
Case
C
2. R11<R21 and R12<R22The following result deals with the number and location of the coexistence equilibriums and with the global behavior of solutions for positive initial conditions.Theorem 6.
Let us consider system (20) and let us assume that ϕi(0,0)>1 for i=1,2.(1)
If there exists only one positive equilibrium that one denotes asE3∗=(e1∗,e2∗), then one has 0<e1∗<R11 and 0<e2∗<R22. In the case that there exist two positive equilibriums E3∗=(e31∗,e32∗) and E4∗=(e41∗,e42∗), then they can be ordered in such a way that 0<e31∗<e41∗<R11 and 0<e42∗<e32∗<R22.(2)
Study of the different cases is as follows:
(i)
CaseA. E1∗ and E2∗ are hyperbolic and unstable. There exists exactly one positive equilibrium E3∗ and it is hyperbolic and attracting. All orbits starting on R∘+2 converge to E3∗.(ii)
CaseB. E1∗ and E2∗ are hyperbolic and attracting. There exists exactly only one positive equilibrium E3∗ which is hyperbolic and a saddle. Moreover E3∗ can not attract any open set and therefore almost all orbits starting in R∘+2 converge to either E1∗ or E2∗.(iii)
CaseC1. E1∗ is hyperbolic and attracting, E2∗ is hyperbolic and unstable, and there can exist zero (case C1a), two (case C1b), or one (nongeneric case) positive equilibriums. Let us deal with the generic cases:
(a)
CaseC1a. All orbits starting on R∘+2 converge to E1∗.(b)
CaseC1b. If one orders the two positive equilibriums E3∗ and E4∗ as indicated in part(1), then no orbits starting in R∘+2 can converge to E2∗. Besides E3∗ is hyperbolic and attracting and E4∗ is hyperbolic and a saddle. Moreover E4∗ can not attract any open set and therefore almost all orbits starting in R∘+2 converge to either E1∗ or E3∗.(iv)
CaseC2. E2∗ is hyperbolic and asymptotically stable, E1∗ is hyperbolic and unstable, and there are no positive equilibriums. All orbits starting on R∘+2 converge to E2∗.Proof.
See Appendix.The analytic expression of the positive equilibrium points is quite complex and we omit it, although we will specify it in a particular case considered in Section3.4.In order to get a qualitative idea of the effect of the disease in the competition dynamics, let us compare the different scenarios regarding the long-term behavior of the classical Leslie-Gower competition model (in the case where there can exist positive equilibriums) with those resulting from Theorem6 in which the effect of a disease in species 1 is taken into account. In the classical Leslie-Gower competition model, we can have (a) competitive exclusion of one of the two species for any initial condition (corresponding to our cases C1a and C2), (b) competitive exclusion of one species depending on the initial condition (corresponding to our case B), and (c) species coexistence for any initial condition (corresponding to our case A). Therefore, the introduction of the disease includes a new scenario which is exclusion of species 2 or species coexistence depending on initial conditions (our case C1b).As we did in Section2, the results on approximate aggregation in [18] together with Theorems 4 and 6 allow one to study the asymptotic behavior of solutions of system (18). Loosely speaking, the results in [18] guarantee that when a solution of the reduced system (20) with initial condition (a1,a2) tends to a hyperbolic equilibrium (x1∗,x2∗) then if k is large enough, the corresponding solution of system (18) (i.e., a solution of (18) with initial condition such that NS1(0)+NI1(0)=a1, N2(0)=a2) tends to an equilibrium which is approximately Tνx1∗,1-νx1∗,x2∗ where T is the mapping, defined by (17), that corresponds to the competition process; that is, (25)Tνx1∗,1-νx1∗,x2∗=bS1νx1∗1+cS1x1∗+cS2x2∗,bI11-νx1∗1+cI1x1∗+cI2x2∗,b2x2∗1+c21x1∗+c22x2∗.In the case in which the disease is endemic, R0>1, the previous expression has the form Tx1∗/R0,1-1/R0x1∗,x2∗.Figure2 shows the basins of attraction of E1∗ and E3∗ for the reduced system together with some solutions of the original system for a choice of parameters corresponding to case C1b. Note in particular that, as predicted by the theory, when the initial condition is such that the solution of the aggregated system tends to E1∗=(x1∗,0), the solution of the original system converges to Tνx1∗,1-νx1∗,0=b1Sνx1∗/(1+cS1x1∗),b1I(1-ν)x1∗/(1+cI1x1∗),0.Figure 2
Basins of attractionB(E1∗) and B(E3∗) of E1∗ and E3∗ for the aggregated system (19) and solutions versus time for the original system (18). The parameter values are k=20, γ=0.9, β=1.125, bS1=10, bI1=20, b2=4, cSS=0.75, cSI=2, cIS=9, cII=14, cS2=6, cI2=1, c2S=0.5, c2I=3, and c22=1, which correspond to case C1b.
(a)
Basins of attractionB(E1∗) and B(E3∗) of E1∗ and E3∗ for system (19) and separatrix curve Γ∗
(b)
Solution of (18) for initial conditions NS1(0)=4, NI1(0)=2, and N2(0)=4, belonging to B(E1∗)
(c)
Solution of (18) for initial conditions NS1(0)=4, NI1(0)=2, and N2(0)=3.5, belonging to B(E3∗)In the next sections we analyze particular cases of the above setting in order to gain more insight into the effect of infection in the dynamics of competition.


### 3.3. Disease Modified Competitive Abilities

In [21] the consequences of parasite-modified competitive abilities of prey are studied in a predator-prey system. They find that they are crucial by showing that they may lead to coexistence of predator and prey, a situation that is impossible otherwise. These findings enhance the importance of the indirect effects of parasite-mediated modifications of host life-history traits beyond the direct effects as, that is, on reproduction.We consider a first particular case in which disease affects only the competitive abilities of infected individuals. Thus we assume that the growth rate of species 1 is unaffected by the disease, that is,b1≔bS1=b. Moreover, we suppose that the competitive abilities of infected individuals in species 1 are worse that those corresponding to the susceptible individuals. Specifically we assume cSS>cSI, c2S>c2I; that is, the interference the susceptible individuals produce either on themselves or on species 2 individuals is larger than the corresponding interference exerted by infected individuals. At the same time we suppose that the rest of intraspecific competition coefficients of species 1 are equal, that is, cII=cIS=cSS.At first glance we could predict that our assumptions will yield a negative impact in the capacity of species 1 to compete against species 2 in the presence of infected individuals. However we will see that, for certain range of parameter values, the combined effect of competition and fast disease dynamics gives rise to scenarios in which the disease endemicity (1/ν=R0>1) improves the outcome of competition for species 1 with respect to the disease-free setting (ν=1).To illustrate this last point we treat the particular example of system (17) with parameters values verifying the above relations. Let b1≔bS1=bI1>1, b2=5, cSS=3, cSI=2.8, cS2=cIS=cII=cI2=c22=1, c2S=2, c2I=1.8, and ν∈(0,1]. Applying Theorem 6 to the corresponding reduced system (19) we show in Figure 3 the asymptotic behavior of solutions of system (17) for a range of values of parameters ν and b1.Figure 3
Asymptotic behavior cases (Theorem6) of the solutions of system (17) for parameters values b1≔bS1=bI1>1, b2=5, cSS=3, cSI=2.8, cS2=cIS=cII=cI2=c22=1, c2S=2, c2I=1.8, and ν∈(0,1].In the first place, notice that all five scenarios shown in Theorem6 can happen. Secondly, we highlight two unexpected behaviors that appear as the basic reproduction number of the disease R0=1/ν increases. The first one corresponds to values of parameter b1 in the approximate range 3.3 to 5. Solutions tend to species 1 exclusion, case C2, in the disease-free situation (ν=1) whereas increasing R0 (decreasing ν) makes case B appear, that is, the exclusion of either species 1 or species 2 depending on the initial sizes of both populations. A second unexpected change of asymptotic behavior appears for b1>5. The tendency to coexistence, case A, in the disease-free scenario is transformed, by increasing R0, into case C1a, exclusion of species 2, sometimes going through C1b, which also entails exclusion of species 2 depending on initial conditions.


### 3.4. Disease Modified Growth Capabilities

We consider now another particular case in which we can strengthen some of the analytical results of Section3.2 and we can extract some interesting quantitative biological information. Let us assume that the disease does not affect the competitive abilities of individuals but it only modifies the reproductive capabilities of the infected species; that is,(26)c11≔cSS=cSI=cIS=cII,c12≔cS2=cI2,c21=c2S=c2I.Then system (19) takes the form (27)N1t+1=rS+rIN1t1+c11N1t+c12N2t,N2t+1=r2N2t1+c21N1t+c22N2twhich is a classical Leslie-Gower model [22] with disease modified growth rates. From the general setting of Section 3.1, the isocline Γ1 now degenerates into a straight line. The isoclines intercepts (24) simplify to(28)R1j=rS+rI-1c1j=bS1ν+bI11-ν-1c1j,R2j=b2-1c2j,j=1,2.Then we can sharpen the results in Theorem 6. In the first place, since the isoclines are straight lines there can be at most one positive equilibrium E3∗=(e1∗,e2∗) which can be computed explicitly as(29)e1∗=c22rS+rI-1-c12r2-1c11c22-c12c21,e2∗=c11r2-1-c21rS+rI-1c11c22-c12c21and, moreover, we have the following.Corollary 7.
Consider system (27). Then, in addition to the results in Theorem 6 one has that case C1b can not happen and therefore in case C1 (R11>R21 and R12>R22) all orbits starting on R∘+2 converge to E1∗.Proof.
The result is immediate taking into account thatΓ1 and Γ2 are now straight lines and therefore conditions R11>R21 and R12>R22 imply that they do not intersect.Let us concentrate in caseA (coexistence for any initial condition). In order to quantify the effect of the disease in the equilibrium population, we set b1=bS1, bI1=αb1, where α>0 is a measure of the effect of the disease on the reproductive capability of infected individuals in species 1. In this way, using (29) we have that the quotient between the equilibrium population of species 1 with (0<ν<1) and without disease (ν=1) is given by(30)c22b1α+1-αν-1-c12b2-1c22b1-1-c12b2-1,while the corresponding quotient for species 2 is(31)c21b1α+1-αν-1-c11b2-1c21b1-1-c11b2-1,so that in particular if α<1 (resp., α>1) the long-term population size of species 1 is reduced (resp., increased) and the opposite happens to species 2.


## 3.1. Model Construction and Reduction

In this section we generalize the setting of the previous one and introduce a second species, which we consider disease free, which competes, according to the well-known Leslie-Gower model [13], with the species which is affected by the disease. Specifically, let us consider two species that we denote by 1 and 2, in competition and a disease defined by system (1) and which affects only species 1. Let NS1 and NI1 represent the number of susceptible and infected individuals of species 1 and let N2 be the amount of individuals of species 2. Then, the slow dynamics is described by(17)NS1t+1=bS1NS1t1+cSSNS1t+cSINI1t+cS2N2t,NI1t+1=bI1NI1t1+cISNS1t+cIINI1t+cI2N2t,N2t+1=b2N2t1+c2SNS1t+c2INI1t+c22N2t,where all growth and competition coefficients are assumed to be positive.As in Section2, we consider that in the time unit of the model a single episode of demographic change happens, defined by (17), followed by a number k of disease infection-recovery cycles corresponding to the SIS model (1) introduced in Section 2. Then, the complete system that takes into account the joint effect of the slow and fast processes takes the form (18)NS1t+1=bS1FSkNS1t,NI1t1+cSSFSkNS1t,NI1t+cSIFIkNS1t,NI1t+cS2N2t,NI1t+1=bI1FIkNS1t,NI1t1+cISFSkNS1t,NI1t+cIIFIkNS1t,NI1t+cI2N2t,N2t+1=b2N2t1+c2SFSkNS1t,NI1t+c2IFIkNS1t,NI1t+c22N2t.If we follow the reduction procedure presented in [15, 17, 18], which essentially consists in adding up the first two equations in (18) and using the approximations NS1≈νNS1+NI1 and NI1≈(1-ν)NS1+NI1, we obtain the reduced system(19)N1t+1=bS1νN1t1+cS1N1t+cS2N2t+bI11-νN1t1+cI1N1t+cI2N2t,N2t+1=b2N2t1+c21N1t+c22N2t,where N1≔NS1+NI1, cS1≔cSSν+cSI1-ν, cI1≔cISν+cII1-ν, and c21≔c2Sν+c2I1-ν.Note that system (19) generalizes the classical Leslie-Gower competition model [13] in the sense that at ν=1 (i.e., the disease-free scenario) system (19) becomes the usual Leslie-Gower competition model.


## 3.2. Model Analysis

In order to carry out the mathematical treatment of system (19) we express it in the form (20)N1t+1,N2t+1=HN1t,N2t,where H is the map defined by(21)Hx1,x2=H1x1,x2,H2x1,x2=ϕ1x1,x2x1,ϕ2x1,x2x2,ϕ1x1,x2≔rS1+cS1x1+cS2x2+rI1+cI1x1+cI2x2,ϕ2x1,x2≔r21+c21x1+c22x2,rS≔bS1ν,rI≔bI11-ν,r2=b2.Note that all the parameters of the model are positive except possibly rI that is nonnegative. As we will see, the numbers (22)ϕ10,0=rS+rI=bS1ν+bI11-ν,ϕ20,0≔r2=b2will play an important role in the dynamics of the system.We note that the functional form of system (20) is a particular case of a model obtained in [19] which studies the dynamics of two spatially distributed populations in competition in which there is fast migration of individuals between the different patches. In that work an analytical study of the system was carried out in the case in which ϕi(0,0)≤1 for at least one value of i=1,2, which essentially corresponds to our Proposition 3 and Theorem 4. In Theorem 6 we provide a full analytical study of the behavior of the orbits of the system in the case ϕi(0,0)>1 for i=1,2, which was only addressed numerically in [19].LetA1=(x1,0):x1>0 and A2=(0,x2):x2>0 be the positive axes. It is immediate to realize that the sets R+2, R∘+2, and Ai, i=1,2 are forward invariant by H. In the sequel, unless otherwise stated we will always assume that we are working on R+2.Let us study the isoclines of the system, that is, the sets defined byx1=H1(x1,x2) and x2=H2(x1,x2). Clearly, besides the positive axes A1 (resp., A2) in which the variable x2 (resp., x1) is constant, the isoclines are the curves Γi defined by ϕi(x1,x2)=1, for i=1,2.Lemma 2.
The isoclinesΓ1 and Γ2 are, respectively, a hyperbola and a straight line. For each i=1,2, Γi intersects R∘+2 if and only if ϕi(0,0)>1 and in that case Γi intersects both the positive axes A1 and A2. Moreover Γ1∩R+2 can be written in the form x2=Φ(x1), x1∈[0,R] for a certain R>0, where Φ is a strictly decreasing convex function.
Except in the degenerate case in whichΓ1 and Γ2 are in fact the same straight line, there can exist at most two positive equilibriums.Proof.
See Appendix.Let us now consider the existence of equilibriums for system (20). Note that E0∗≔(0,0) is an equilibrium point for all values of the parameters. Using Lemma 2 we conclude the follwoing:(i)
For eachi=1,2, there exists a semitrivial equilibrium point Ei∗ on the positive axis Ai if and only if ϕi(0,0)>1. In that case the semitrivial equilibrium is unique.(ii)
A necessary condition for the existence of a positive equilibrium is thatϕi(0,0)>1 for both i=1,2, and in that case there can exist at most two such equilibriums.In what follows we will writex1,x2≤x1′,x2′ (resp., x1,x2<x1′,x2′) to denote that x1≤x1′ and x2≤x2′ (resp., x1<x1′ and x2<x2′). Similarly, we will write x1,x2≤Kx1′,x2′ (resp., x1,x2<Kx1′,x2′) to denote that x1≤x1′ and x2≥x2′ (resp., x1<x1′ and x2>x2′).Proposition 3.
Let us consider system (20).(a)
All solutions inR+2 are forward bounded. More specifically, (23)HR+2⊂S≔0,rScS1+rIcI1×0,r2c22.(b)
H is strongly competitive in R+2; that is, if x,x′∈R+2 are distinct points with x≤Kx′ it follows that H(x)<KH(x′) [20].(c)
All orbits inR+2 are eventually componentwise monotone; that is, for each N1(0),N2(0)∈R+2, the corresponding solution N1(t),N2(t) verifies that Ni(t) is eventually monotone for each i=1,2. Moreover, all orbits tend to an equilibrium as t→∞.Proof.
See Appendix.The next result analyzes the behavior of solutions of system (20) except in the case in which ϕi(0,0)>1, i=1,2.Theorem 4.
Let us consider system (20).(a)
For eachi=1,2, if ϕi(0,0)≤1 then for any initial value on R+2 species i tends to extinction as t→∞.(b)
Ifϕi(0,0)≤1 for i=1,2, all orbits in R+2 tend to E0∗, and if the inequalities are strict then E0∗ is hyperbolic. If ϕi(0,0)>1 for i=1,2, then E0∗ is a repeller and therefore no orbit can converge to E0∗.(c)
Ifϕ1(0,0)>1 and ϕ2(0,0)≤1 then(1) all orbits with N1(0)=0 tend to E0∗ and(2) all orbits with N1(0)>0 tend to E1∗.(d)
Ifϕ2(0,0)>1 and ϕ1(0,0)≤1 then(1) all orbits with N2(0)=0 tend to E0∗ and(2) all orbits with N2(0)>0 tend to E2∗.Proof.
See Appendix.Let us now consider the case in whichϕi(0,0)>1 for i=1,2. In the first place we study the local stability of the semitrivial equilibriums and the behavior of orbits on the positive axes.Proposition 5.
Let us consider system (20) and let us assume that ϕi(0,0)>1 for i=1,2.(a)
All orbits starting on the positivei-axis Ai converge to Ei∗, i=1,2.(b)
Leti=1,2 be fixed. If ϕj(Ei∗)<1, j≠i, then Ei∗ is hyperbolic and attracting. If ϕj(Ei∗)>1, j≠i, then Ei∗ is hyperbolic and unstable.Proof.
(a) Leti=1,2 be fixed. We know that Ai is invariant for H and then any orbit corresponding to an initial condition on Ai must converge to either E0∗ or Ei∗. From Theorem 4 part (b) E0∗ is a repeller and therefore the result follows.
(b) It follows from the usual analysis of the eigenvalues of the corresponding Jacobian matrix. Standard calculations lead to the results bearing in mind thatϕi(Ei∗)=1.In order to study the behavior of solutions for positive initial conditions, we will consider different cases based on the relative position of the intercepts of the isoclines. To this end, we defineRij as the j-intercept of Γi, i,j=1,2. It is immediate to check that Rij is given by(24)R1j=12cSjcIjαj+αj2+4cSjcIjrS+rI-1,j=1,2,R2j=r2-1c2j,j=1,2,where αj≔rS(cIj-1)+rI(cSj-1), j=1,2.Now we distinguish the following scenarios:(i)
CaseA: R11<R21 and R12>R22.(ii)
CaseB: R11>R21 and R12<R22.(iii)
CaseC1: R11>R21 and R12>R22.(iv)
CaseC2: R11<R21 and R12<R22.Taking into account Lemma2, the isoclines divide R∘+2 in a finite number of open connected and disjoint sets Uk, k=1,…,q, in which xi is either strictly increasing or strictly decreasing for each i=1,2. To describe the kind of monotonicity in each region we will use arrows, in such a way that, for example, the situation in which x1 decreases and x2 increases corresponding to a horizontal arrow pointing to the left and a vertical arrow pointing up. Figure 1 shows the different configurations. Note that in case C1 we can distinguish two generic cases depending on whether the two isoclines do not intersect (case C1a) or intersect at two points (case C1b).Figure 1
Different configurations of isoclines and equilibrium points of system (20) when ϕi(0,0)>1 for i=1,2, in terms of the relative position of the intercepts of isoclines, Rij (24).
(a)
Case A. R11<R21 and R12>R22
(b)
Case B. R11>R21 and R12<R22
(c)
Case
C
1
a. R11>R21 and R12>R22 and no positive equilibrium
(d)
Case
C
1
b. R11>R21 and R12>R22 and two positive equilibriums
(e)
Case
C
2. R11<R21 and R12<R22The following result deals with the number and location of the coexistence equilibriums and with the global behavior of solutions for positive initial conditions.Theorem 6.
Let us consider system (20) and let us assume that ϕi(0,0)>1 for i=1,2.(1)
If there exists only one positive equilibrium that one denotes asE3∗=(e1∗,e2∗), then one has 0<e1∗<R11 and 0<e2∗<R22. In the case that there exist two positive equilibriums E3∗=(e31∗,e32∗) and E4∗=(e41∗,e42∗), then they can be ordered in such a way that 0<e31∗<e41∗<R11 and 0<e42∗<e32∗<R22.(2)
Study of the different cases is as follows:
(i)
CaseA. E1∗ and E2∗ are hyperbolic and unstable. There exists exactly one positive equilibrium E3∗ and it is hyperbolic and attracting. All orbits starting on R∘+2 converge to E3∗.(ii)
CaseB. E1∗ and E2∗ are hyperbolic and attracting. There exists exactly only one positive equilibrium E3∗ which is hyperbolic and a saddle. Moreover E3∗ can not attract any open set and therefore almost all orbits starting in R∘+2 converge to either E1∗ or E2∗.(iii)
CaseC1. E1∗ is hyperbolic and attracting, E2∗ is hyperbolic and unstable, and there can exist zero (case C1a), two (case C1b), or one (nongeneric case) positive equilibriums. Let us deal with the generic cases:
(a)
CaseC1a. All orbits starting on R∘+2 converge to E1∗.(b)
CaseC1b. If one orders the two positive equilibriums E3∗ and E4∗ as indicated in part(1), then no orbits starting in R∘+2 can converge to E2∗. Besides E3∗ is hyperbolic and attracting and E4∗ is hyperbolic and a saddle. Moreover E4∗ can not attract any open set and therefore almost all orbits starting in R∘+2 converge to either E1∗ or E3∗.(iv)
CaseC2. E2∗ is hyperbolic and asymptotically stable, E1∗ is hyperbolic and unstable, and there are no positive equilibriums. All orbits starting on R∘+2 converge to E2∗.Proof.
See Appendix.The analytic expression of the positive equilibrium points is quite complex and we omit it, although we will specify it in a particular case considered in Section3.4.In order to get a qualitative idea of the effect of the disease in the competition dynamics, let us compare the different scenarios regarding the long-term behavior of the classical Leslie-Gower competition model (in the case where there can exist positive equilibriums) with those resulting from Theorem6 in which the effect of a disease in species 1 is taken into account. In the classical Leslie-Gower competition model, we can have (a) competitive exclusion of one of the two species for any initial condition (corresponding to our cases C1a and C2), (b) competitive exclusion of one species depending on the initial condition (corresponding to our case B), and (c) species coexistence for any initial condition (corresponding to our case A). Therefore, the introduction of the disease includes a new scenario which is exclusion of species 2 or species coexistence depending on initial conditions (our case C1b).As we did in Section2, the results on approximate aggregation in [18] together with Theorems 4 and 6 allow one to study the asymptotic behavior of solutions of system (18). Loosely speaking, the results in [18] guarantee that when a solution of the reduced system (20) with initial condition (a1,a2) tends to a hyperbolic equilibrium (x1∗,x2∗) then if k is large enough, the corresponding solution of system (18) (i.e., a solution of (18) with initial condition such that NS1(0)+NI1(0)=a1, N2(0)=a2) tends to an equilibrium which is approximately Tνx1∗,1-νx1∗,x2∗ where T is the mapping, defined by (17), that corresponds to the competition process; that is, (25)Tνx1∗,1-νx1∗,x2∗=bS1νx1∗1+cS1x1∗+cS2x2∗,bI11-νx1∗1+cI1x1∗+cI2x2∗,b2x2∗1+c21x1∗+c22x2∗.In the case in which the disease is endemic, R0>1, the previous expression has the form Tx1∗/R0,1-1/R0x1∗,x2∗.Figure2 shows the basins of attraction of E1∗ and E3∗ for the reduced system together with some solutions of the original system for a choice of parameters corresponding to case C1b. Note in particular that, as predicted by the theory, when the initial condition is such that the solution of the aggregated system tends to E1∗=(x1∗,0), the solution of the original system converges to Tνx1∗,1-νx1∗,0=b1Sνx1∗/(1+cS1x1∗),b1I(1-ν)x1∗/(1+cI1x1∗),0.Figure 2
Basins of attractionB(E1∗) and B(E3∗) of E1∗ and E3∗ for the aggregated system (19) and solutions versus time for the original system (18). The parameter values are k=20, γ=0.9, β=1.125, bS1=10, bI1=20, b2=4, cSS=0.75, cSI=2, cIS=9, cII=14, cS2=6, cI2=1, c2S=0.5, c2I=3, and c22=1, which correspond to case C1b.
(a)
Basins of attractionB(E1∗) and B(E3∗) of E1∗ and E3∗ for system (19) and separatrix curve Γ∗
(b)
Solution of (18) for initial conditions NS1(0)=4, NI1(0)=2, and N2(0)=4, belonging to B(E1∗)
(c)
Solution of (18) for initial conditions NS1(0)=4, NI1(0)=2, and N2(0)=3.5, belonging to B(E3∗)In the next sections we analyze particular cases of the above setting in order to gain more insight into the effect of infection in the dynamics of competition.


## 3.3. Disease Modified Competitive Abilities

In [21] the consequences of parasite-modified competitive abilities of prey are studied in a predator-prey system. They find that they are crucial by showing that they may lead to coexistence of predator and prey, a situation that is impossible otherwise. These findings enhance the importance of the indirect effects of parasite-mediated modifications of host life-history traits beyond the direct effects as, that is, on reproduction.We consider a first particular case in which disease affects only the competitive abilities of infected individuals. Thus we assume that the growth rate of species 1 is unaffected by the disease, that is,b1≔bS1=b. Moreover, we suppose that the competitive abilities of infected individuals in species 1 are worse that those corresponding to the susceptible individuals. Specifically we assume cSS>cSI, c2S>c2I; that is, the interference the susceptible individuals produce either on themselves or on species 2 individuals is larger than the corresponding interference exerted by infected individuals. At the same time we suppose that the rest of intraspecific competition coefficients of species 1 are equal, that is, cII=cIS=cSS.At first glance we could predict that our assumptions will yield a negative impact in the capacity of species 1 to compete against species 2 in the presence of infected individuals. However we will see that, for certain range of parameter values, the combined effect of competition and fast disease dynamics gives rise to scenarios in which the disease endemicity (1/ν=R0>1) improves the outcome of competition for species 1 with respect to the disease-free setting (ν=1).To illustrate this last point we treat the particular example of system (17) with parameters values verifying the above relations. Let b1≔bS1=bI1>1, b2=5, cSS=3, cSI=2.8, cS2=cIS=cII=cI2=c22=1, c2S=2, c2I=1.8, and ν∈(0,1]. Applying Theorem 6 to the corresponding reduced system (19) we show in Figure 3 the asymptotic behavior of solutions of system (17) for a range of values of parameters ν and b1.Figure 3
Asymptotic behavior cases (Theorem6) of the solutions of system (17) for parameters values b1≔bS1=bI1>1, b2=5, cSS=3, cSI=2.8, cS2=cIS=cII=cI2=c22=1, c2S=2, c2I=1.8, and ν∈(0,1].In the first place, notice that all five scenarios shown in Theorem6 can happen. Secondly, we highlight two unexpected behaviors that appear as the basic reproduction number of the disease R0=1/ν increases. The first one corresponds to values of parameter b1 in the approximate range 3.3 to 5. Solutions tend to species 1 exclusion, case C2, in the disease-free situation (ν=1) whereas increasing R0 (decreasing ν) makes case B appear, that is, the exclusion of either species 1 or species 2 depending on the initial sizes of both populations. A second unexpected change of asymptotic behavior appears for b1>5. The tendency to coexistence, case A, in the disease-free scenario is transformed, by increasing R0, into case C1a, exclusion of species 2, sometimes going through C1b, which also entails exclusion of species 2 depending on initial conditions.


## 3.4. Disease Modified Growth Capabilities

We consider now another particular case in which we can strengthen some of the analytical results of Section3.2 and we can extract some interesting quantitative biological information. Let us assume that the disease does not affect the competitive abilities of individuals but it only modifies the reproductive capabilities of the infected species; that is,(26)c11≔cSS=cSI=cIS=cII,c12≔cS2=cI2,c21=c2S=c2I.Then system (19) takes the form (27)N1t+1=rS+rIN1t1+c11N1t+c12N2t,N2t+1=r2N2t1+c21N1t+c22N2twhich is a classical Leslie-Gower model [22] with disease modified growth rates. From the general setting of Section 3.1, the isocline Γ1 now degenerates into a straight line. The isoclines intercepts (24) simplify to(28)R1j=rS+rI-1c1j=bS1ν+bI11-ν-1c1j,R2j=b2-1c2j,j=1,2.Then we can sharpen the results in Theorem 6. In the first place, since the isoclines are straight lines there can be at most one positive equilibrium E3∗=(e1∗,e2∗) which can be computed explicitly as(29)e1∗=c22rS+rI-1-c12r2-1c11c22-c12c21,e2∗=c11r2-1-c21rS+rI-1c11c22-c12c21and, moreover, we have the following.Corollary 7.
Consider system (27). Then, in addition to the results in Theorem 6 one has that case C1b can not happen and therefore in case C1 (R11>R21 and R12>R22) all orbits starting on R∘+2 converge to E1∗.Proof.
The result is immediate taking into account thatΓ1 and Γ2 are now straight lines and therefore conditions R11>R21 and R12>R22 imply that they do not intersect.Let us concentrate in caseA (coexistence for any initial condition). In order to quantify the effect of the disease in the equilibrium population, we set b1=bS1, bI1=αb1, where α>0 is a measure of the effect of the disease on the reproductive capability of infected individuals in species 1. In this way, using (29) we have that the quotient between the equilibrium population of species 1 with (0<ν<1) and without disease (ν=1) is given by(30)c22b1α+1-αν-1-c12b2-1c22b1-1-c12b2-1,while the corresponding quotient for species 2 is(31)c21b1α+1-αν-1-c11b2-1c21b1-1-c11b2-1,so that in particular if α<1 (resp., α>1) the long-term population size of species 1 is reduced (resp., increased) and the opposite happens to species 2.


## 4. Conclusion

In this work we have built up an ecoepidemic model in which disease dynamics is fast with respect to competition-demography and have carried out its mathematical analysis, which reveals a number of possible long-term scenarios for the populations.From an applied point of view, biological controls as predators or competitors, along with disease-induced population size reduction, are often used to control harmful species. Some key questions are whether a given control strategy will be effective or whether it hides unexpected effects (as the well-known hydra effect [23]). In this direction we have obtained threshold values for species persistence/extinction (as in Theorems 1 and 4 or Proposition 5) and expressions (like (16), (30), and (31)) that provide an estimation of the impact on the community of a disease outbreak. Our analysis also reveals that, for certain values of the parameters, unexpected outcomes as those described in Section 3.3 may appear. There we have shown that there are situations in which as the basic reproductive number R0 increases, the infected species improves as a competitor even though the interspecific competitive pressure exercised by infected individuals is lower than that exercised by susceptible individuals.The theoretical results obtained in this work can encourage laboratory experiments to validate some of the unexpected results predicted by the model as well as modeling work in order to find plausible explanations for these scenarios. Our findings also suggest possible future lines of work, for example, the study of competition models in which both species are infected by either a shared disease or, alternatively, different diseases.



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*Source: 5310837-2017-09-20.xml*