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.
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 [
In this work we focus on ecoepidemic competition models that have attracted great attention [
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 of
From the mathematical point of view, complete ecoepidemic competition models are known for the difficulty of their analytical study (see [
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 the
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 [
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 [
This work is organized as follows: in Section
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 number
Time in the slow time unit is denoted by
The disease dynamics is defined by means of the discrete time SIS epidemic model studied in [
We assume that transmission is frequency-dependent, with
The disease dynamics keeps constant the population size
Henceforth we assume that the following inequalities hold:
The asymptotic behavior of the solutions of the discrete system defined by map
If
In the following we will restrict to the case where (
To build up our demographic model where we have to take into account susceptible and infected individuals, we start from the Beverton-Holt model [ If If
Note that this corresponds to a logistic behavior with carrying capacity
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,
To build up the complete model combining the demographic and the disease processes we compose the
System (
We can write (
One considers equation ( If If
See Th. A.4 in [
Making use of the last theorem and the results on approximate aggregation in [ If If
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 (
It is straightforward to check that
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
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 [
As in Section
Note that system (
In order to carry out the mathematical treatment of system (
We note that the functional form of system (
Let
Let us study the isoclines of the system, that is, the sets defined by
The isoclines
Except in the degenerate case in which
See Appendix.
Let us now consider the existence of equilibriums for system ( For each A necessary condition for the existence of a positive equilibrium is that
In what follows we will write
Let us consider system ( All solutions in All orbits in
See Appendix.
The next result analyzes the behavior of solutions of system (
Let us consider system ( For each If If If
See Appendix.
Let us now consider the case in which
Let us consider system ( All orbits starting on the positive Let
(a) Let
(b) It follows from the usual analysis of the eigenvalues of the corresponding Jacobian matrix. Standard calculations lead to the results bearing in mind that
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 define
Now we distinguish the following scenarios: Case Case Case Case
Taking into account Lemma
Different configurations of isoclines and equilibrium points of system (
The following result deals with the number and location of the coexistence equilibriums and with the global behavior of solutions for positive initial conditions.
Let us consider system ( If there exists only one positive equilibrium that one denotes as Study of the different cases is as follows: Case Case Case Case Case Case
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 Section
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 Theorem
As we did in Section
Figure
Basins of attraction
Basins of attraction
Solution of (
Solution of (
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.
In [
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,
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 (
To illustrate this last point we treat the particular example of system (
Asymptotic behavior cases (Theorem
In the first place, notice that all five scenarios shown in Theorem
We consider now another particular case in which we can strengthen some of the analytical results of Section
Consider system (
The result is immediate taking into account that
Let us concentrate in case
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 [
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.
Clearly
Clearly, if
The last part of the result follows taking into account the fact that a branch of a hyperbola can intersect a straight line in at most 2 points.
The proof of (a) is straightforward. Regarding (b), it is immediate to check that, for
(c) In the first place we prove that It is immediate that To prove that Direct calculations prove that The last hypothesis to prove is that
Now, using (a) and (b) and property (O+) we can apply Theorem
(a) Let
(b) The first part is trivial from (a) and the rest of the result is easy to check using the linearization of
(c) If
(d) The proof of the result is analogous to that of (c).
(a) For any initial condition
(b) Let
(c) Let
(a) The result is a direct consequence of the fact that orbits are eventually componentwise monotone.
(b) The result is a direct consequence of the fact that
(c) The result is a direct consequence of the fact that property (O+) (part (c) of Proposition
Let
Let
In the proof of Proposition
(1) The bounds for the components of the positive equilibrium or equilibriums are a trivial consequence of the fact that
(2) Throughout the proof we will use implicitly the fact that
Clearly
Using Lemma
Clearly
Let us now show that
In scenario (i) we have already shown that the corresponding orbit converges to
In order to prove that there are points different from
The proof that
The authors declare that they have no conflicts of interest.
The authors are supported by Ministerio de Economía y Competitividad (Spain), Project MTM2014-56022-C2-1-P.