Periodic orbits for periodic eco-epidemiological systems with infected prey

We address the existence of periodic orbits for periodic eco-epidemiological system with disease in the prey. To do it, we consider three main steps. Firstly we study a one parameter family of systems and obtain uniform bounds for the components of any periodic solution of these systems. Next, we make a suitable change of variables in our family of systems to establish the setting where we are able to apply Mawhin's continuation Theorem. Finally, we use Mawhin's continuation Theorem to obtain our result. Later on, we present two examples that include previous results in the literature and some numerical simulations to illustrate our results.


Introduction
Eco-epidemiological models are ecological models that include infected compartments. In many situations, these models describe more accurately the real ecological system than models where the disease is not taken into account.
There is already a large number of works concerning eco-epidemiological models. To mention just a few recent works, we refer [4] where a mathematical study on disease persistence and extinction is carried out; [5] where the authors study the global stability of a delayed eco-epidemiological model with holling type III functional response, and [2] where an eco-epidemiological model with harvesting is considered.
One of the main concerns when studying eco-epidemiological models is to determine conditions under which one can predict if the disease persists or dies out. In mathematical epidemiology, these conditions are usually given in terms of the so called basic reproduction ratio R 0 , defined in [8] for autonomous systems as the spectral radius of the next generation matrix.
In [7], R 0 was introduced for the periodic models, and later on, in [10], the definition of R 0 was adapted to the study of periodic patchy models. In the recent article [6] the theory in [10] was used in the study of persistence of the predator in a general periodic predator-prey models.
When persistence is guaranteed, the obtention of conditions that assure the existence of periodic orbits for periodic eco-epidemiological models is an important issue in the deepening of the description of these models since these orbits correspond to situations where possibly there is some equilibrium in the described ecological system, reflected in the fact that the behaviour of the theoretical model is the same over the years. In [3] it was proved that there is an endemic periodic orbit for the periodic version of the model considered in [11] when the infected prey is permanent and some additional conditions are fulfilled, partially giving a positive answer to a conjecture in this last paper.
The models in [11] and [3] assume that there is no predation on uninfected preys. In spite of that, this assumption is not suitable for the description of many ecoepidemiological models. The main purpose of this paper is to present some results on the existence of an endemic periodic orbit for periodic eco-epidemiological systems with disease in the prey that generalize the systems in [11] and [3] by including in the model a general function corresponding to the predation of uninfected preys. The proof of our result relies on Mawhin's continuation theorem. Following the approach in [3], we begin by locating the components of possible periodic orbits for the one parameter family of systems that arise in Mawhin's result, allowing us to check that the conditions of that theorem are fulfilled. Although the main steps in our proof correspond to the ones in [3], several additional nontrivial arguments are needed in our case. Additionally, there is also a substantial difference between our approach and the one in [11,3]. In fact, we take as a departure point some prescribed behaviour of the uninfected subsystem, corresponding to the dynamics of preys and predators in the absence of disease: we will assume in this work that we have global asymptotic stability of solutions of some special perturbations of the bidimensional predator-prey system (the system obtained by letting I = 0 in the first and third equations in (1)). Thus, when applying our results to particular situations, one must verify that the underlying uninfected subsystem satisfies our assumptions. On the other hand, our approach allows us to construct an eco-epidemiological model from a previously studied predator-prey model (the uninfected subsystem) that satisfies our assumptions. This approach has the advantage of highlighting the link between the dynamics of the eco-epidemiological model and the dynamics of the predator-prey model used in its construction.

A general eco-epidemiological model with disease in prey
As a generalization of the model considered in [3], a periodic version of the general non-autonomous model introduced in [11], we consider the following periodic ecoepidemiological model: where S, I and P correspond, respectively, to the susceptible prey, infected prey and predator, Λ(t) is the recruited rate of the prey population, µ(t) is the natural death rate of the prey population, a(t) predation rate of susceptible prey, β(t) is the incidence rate, η(t) is the predation rate of infected prey, c(t) is the death rate in the infective class (c(t) µ(t)), γ(t) is the rate converting susceptible prey into predator (biomass transfer), θ(t) is the rate of converting infected prey into predator, r(t) and b(t) are parameters related the vital dynamics of the predator population that is assumed to follow a logistic law and includes the intra-specific competition between predators. It is assumed that only susceptible preys S are capable of reproducing, i.e, the infected prey is removed by death (including natural and disease-related death) or by predation before having the possibility of reproducing.
Given a ω-periodic function f we will use throughout the paper the notations ω ω 0 f (s) ds. We will assume the following structural hypothesis concerning the parameter functions and the function f appearing in our model: S1) The real valued functions Λ, µ, β, η, c, γ, r, θ and b are periodic with period ω, nonnegative and continuous; S2) Function f is nonnegative and continuous; To formulate our next assumptions we need to consider two auxiliary equations and one auxiliary system. First, for each λ ∈ (0, 1], we need to consider the following equations: and Note that, if we identify x with the susceptible prey population, equation 2 gives the behavior of the susceptible preys in the absence of infected preys and predator and identifying z with the predator population, equation 3 gives the behavior of the predator in the absence of preys. Equations (2) and (3) have a well known behavior, given in the following lemmas: Lemma 1 (Lemma 1 in [11]). For each λ ∈ (0, 1] there is a unique ω-periodic solution of equation (2), x * λ (t), that is globally asymptotically stable in R + . Lemma 2 (Lemma 2 in [11]). For each λ ∈ (0, 1] there is a unique ω-periodic solution of equation (3), z * λ (t), that is globally asymptotically stable in R + . For each λ ∈ (0, 1], we also need to consider the next family of systems, which correspond to behavior of the preys and predators in the absence of infected preys (system (1) with I = 0, S = x and P = z): We now make our last structural assumption on system (1): S9) For each λ ∈ (0, 1] and each ε 1 , ε 2 0 sufficiently small, system (4) has a unique ω-periodic solution, ( Denoting x * λ = x * λ,0,0 and z * λ = z * λ,0,0 , we introduce the numbers

Main result
We now present our main result.
then system (1) possesses an endemic periodic orbit of period ω.
Our proof relies on an application of Mawhin's continuation theorem. We will proceed in several steps. Firstly, in subsection 3.1, we consider a one parameter family of systems and obtain uniform bounds for the components of any periodic solution of these systems. Next, in subsection 3.2 we make a suitable change of variables in our family of systems to establish the setting where we will apply Mawhin's continuation Theorem. Finally, in subsection 3.3, we use Mawhin's continuation Theorem to obtain our result.
3.1. Uniform Persistence for the periodic orbits of a one parameter family of systems. In this section, to obtain uniform bounds for the components of any periodic solution of the family of systems that we can obtain multiplying the right hand side of (1) by λ ∈ (0, 1], we need to consider the auxiliary systems: We will consider separately each of the several components of any periodic orbit. Lemma 3. Let x * λ (t) be the unique solution of (2). There is L 1 > 0 such that, for any λ ∈ (0, 1] and any periodic solution (S λ (t), I λ (t), P λ (t)) of (7) with initial Proof. Let (S λ (t), I λ (t), P λ (t)) be some periodic solution of (7) with initial condi- we have, by the first and second equations of (7), Since, by Lemma 1, equation (2) has a unique periodic orbit, x * λ (t), that is globally asymptotically stable, we conclude that S λ (t)+I λ (t) x * λ (t) for all t ∈ R. Comparing equation (2) with equation x ′ = λΛ u − λµ ℓ x, we conclude that x * λ (t) Λ u /µ ℓ . Using conditions S3) and S4), by the third equation of (7), we have where function Θ is given by Thus, comparing with equation (3) and using Lemma 2, we get P λ (t) P * λ (t) Θ u /b ℓ . Using the bound obtained above, since −β(t)S λ (t) −β(t)x * λ (t), we have, by conditions S3), S4) and S8), be the unique solution of (3). There is L 2 > 0 such that, for any λ ∈ (0, 1] and any periodic solution , P λ (t)) be any periodic solution of (7) with initial conditions S λ (t 0 ) = S 0 > 0, I λ (t 0 ) = I 0 > 0 and P λ (t 0 ) = P 0 > 0. We have Comparing the previous inequality with equation (3) and using Lemma 2, we get P λ (t) z * λ (t). Moreover, comparing equation (3) with equation z ′ = λ(r ℓ − b u z)z we conclude that z * λ (t) r ℓ /b u . Using the computations in proof of the previous lemma, we have P λ (t) L 1 and we take L 2 = L 1 .
Thus, using condition S9), we have where ϕ is a nonegative function such that ϕ(ε) → 0 as ε → 0 (notice that, by continuity, we can assume that ϕ is independent of λ and, by periodicity of the parameter functions, it is independent of t).
To define the operators in Mawhin's theorem (see appendix A), we need to consider the Banach spaces (X, · ) and (Z, · ) where X and Z are the space of ω-periodic continuous functions u : R → R 3 : Next, we consider the linear map L : X ∩ C 1 (R, R 3 ) → Z given by and the map N : (13) In the following lemma we show that the linear map in (12) is a Fredholm mapping of index zero Lemma 6. The linear map L in (12) is a Fredholm mapping of index zero.
Proof. We have and thus ker L can be identified with R 3 . Therefore dim ker L = 3. On the other hand and any z ∈ Z can be written as z =z + α, where α = (α 1 , α 2 , α 3 ) ∈ R 3 and z ∈ Im L. Thus the complementary space of Im L consists of the constant functions. Thus, the complementary space has dimension 3 and therefore codim Im L = 3. Given any sequence (z n ) in Im L such that we have, for i = 1, 2, 3 (note that z ∈ Z since Z is a Banach space and thus it is integrable in [0, ω] since it is continuous in that interval), Thus, z ∈ Im L and we conclude that Im L is closed in Z. Thus L is a Fredholm mapping of index zero.
Consider the projectors P : X → X and Q : Z → Z given by Note that Im P = ker L and that ker Q = Im(I − Q) = Im L.
Consider the generalized inverse of L, K : Im L → D ∩ ker P , given by where Proof. Let U ⊆ X be an open bounded set and U its closure in X. Then, there is M > 0 such that, for any u = (u1, u2, u3) ∈ U , we have that |ui(t)| M , i = 1, 2, 3. Letting  QN u = ((QN )1u, (QN )2u, (QN ) Let B ⊂ X be a bounded set. Note that the boundedness of B implies that there is M such that |ui| < M , for all i = 1, 2, 3, and all u = (u1, u2, u3) ∈ B. It is immediate that and similarly and By (14), (15) and (16), we conclude that {K(I − Q)N u : u ∈ B} is equicontinuous. Therefore, by Ascoli-Arzela's theorem, K(I − Q)N (B) is relatively compact. Thus the operator K(I − Q)N is compact. We conclude that N is L-compact in the closure of any bounded set contained in X.

3.3.
Application of Mawhin's continuation theorem. In this section we will construct the set where, applying Mahwin's continuation theorem, we will find the periodic orbit in the statement of our result.
Consider the system of algebraic equations: By the second and third equations we get e u 1 = η e u 3 +c β and e u 2 = − γa θη f η e u 3 +c β , e u 3 + b θη e u 3 − r θη .
Therefore, using the first equation, we get Consider the function G : [r/b, +∞[→ R (notice that, by the third equation in (17), we have e u 3 r/b), given by and observe that function G1 is decreasing and functions G2 and G3 are increasing. Thus, Consequently G is a decreasing function and equation (17) has, at most, one solution.
It is easy to verify that lim x→+∞ G(x) = −∞ and, by the hypothesis in our theorem Thus we conclude that there is a unique solution of equation (17). Denote this solution by p * (t) = (p * 1 , p * 2 , p * 3 ). By Lemmas 3, 4 and 5, there is a constant M0 > 0 such that u λ (t) < M0, for any t ∈ [0, ω] and any periodic solution u λ (t) of (7). Let Conditions M1. and M2. in Mawhin's continuation theorem (see appendix A) are fulfilled in the set U defined in (18). Using the notation v = (e p * 1 , e p * 3 ), the Jacobian matrix of the vector field corresponding to (17) computed in (p * 1 , p * 2 , p * 3 ) is Since we are assuming that βγa − θηa 0, we have Since f (S, P ) = S, conditions S2) to S6) are trivially satisfied. Conditions S1) and S7) are assumed and S8) is satisfied with K = α = 1. Notice additionally that system (4) becomes in our context System (23) has two equilibriums: E1 = (Λ/(µ + ε1), 0) and where V = µ + ε1 + a(r + ε2)/b. It is easy to check that E2 is locally attractive and that E1 is a saddle point whose stable manifold coincides with the x-axis. If 0 < α < (r + ε2)/b then, in the line z = α the flow points upward. Additionally, if Λ < µ(µ + ε1)/a, in the line x = µ/a the flow points to the left and the x-coordinate of E1 is less than µ/a. Thus the region R = {(x, z) ∈ R 2 : 0 x µ/a ∧ z α} is positively invariant. Since the divergence of the vector field is given by −µ − ε1 + ε2 − (a + 2b)z + γax, we conclude that it is null on the line z = −µ−ε 1 +ε 2 a+2b + γa a+2b x. Thus the divergence of the vector field doesn't change sign on the region R and this forbids the existence of a peridic orbit on R. There is also no periodic orbit on (R + 0 ) 2 \ R since there is no additional equilibrium in (R + 0 ) 2 . Since E2 is locally asymptotically stable, there is no homoclinic orbit conecting E2 to itself. Therefore, the ω-limit of any orbit in (R 2 ) + must be the equilibrium point E2 and the global asymptotic stability of (23) for sufficiently small ε1, ε2 > 0 follows. We conclude that condition S9) holds.
Theorem 2 (Mawhin's continuation theorem). Let X and Z be Banach spaces and let U ⊂ X be an open set. Assume that L : D ⊆ X → Z is a Fredholm mapping of index zero and let N : X → Z be L-compact on U . Additionally, assume that M1) for each λ ∈ (0, 1) and x ∈ ∂U ∩ D we have Lx = λN x; M2) for each x ∈ ∂U ∩ ker L we have QN x = 0; M3) deg(IQN , U ∩ ker L, 0) = 0. Then the operator equation Lx = N x has at least one solution in D ∩ U .