A periodic and diffusive predator-prey model with disease in the prey

In this paper, we are concerned with a time periodic and diffusivepredator-prey model with disease transmission in the prey. Firstwe consider a \begin{document}$ SI $\end{document} model when the predator species is absent. Byintroducing the basic reproduction number for the \begin{document}$ SI $\end{document} model, weshow the sufficient conditions for the persistence and extinctionof the disease. When the presence of the predator is taken intoaccount, a number of sufficient conditions for the co-existence ofthe prey and predator species, the global extinction of predatorspecies and the global extinction of both the prey and predatorspecies are given.


1.
Introduction. Eco-epidemiology models have become important and efficient tools in analyzing the spread of infectious disease between the interacting species. Hadeler and Freedman [7] developed and analysed a predator-prey model, where both species were involved with parasitism. In order to account for the influence of disease transmission on two interacting species, Venturino [24,25,26] proposed and investigated several epidemiology models coupled with Lotka-Volterra model. Since then many elegant works have been dedicated to this direction, see [1,3,4,6,12,16,17,21,29,30,33,35] and references therein.
It is well known that spatial heterogeneity of the population habitat environment and spatial-temporal movement of individuals play significant roles in the dynamics of infectious disease (see, e.g., [20,28]) and the evolution of interacting species (see, e.g. [2]). And not only that, the spread of disease and the growth of population individuals are influenced dramatically on seasonal variation (see e.g. [9,5,18,32]).
In this paper, we shall incorporate the spatial-temporal factors into the general ecoepidemiology model with disease in the prey: x)∇S(t, x)] + a(t, x)S(t, x) −b(t, x)S(t, x) (S(t, x) + I(t, x)) − β(t, x)S(t, x)I(t, x), x)∇I(t, x)] − c(t, x)I(t, x) + β(t, x)S(t, x)I(t, x) −f 1 (t, x, I(t, x), W (t, x)), x, I(t, x), W (t, x)). (1) In this model, we have two populations: the prey (S + I) and the predator W. Assume that the population habitat Ω is a bounded domain in R n (n ≥ 1) with a smooth boundary ∂Ω. We further make the following assumptions: (H1): In the absence of disease, the growth of prey population obey the logistic type law. Let N (t, x) be the density of prey population at time t and location x. Then N (t, x) satisfies subject to no flux boundary condition where ∇ · [D N (t, x)∇N (t, x)] denotes the divergence of D N (t, x)∇N (t, x), D N (t, x) is the diffusion rate at time t and location x, n is the outward normal to ∂Ω. (H2): In the presence of disease, we assume that the total prey population N is composed of susceptible class S and infected class I. Then, the density of total prey population is N (t, x) = S(t, x) + I(t, x) at time t and location x. (H3): Assume that only susceptible prey S is responsible for reproducing, i.e., the infected prey I is removed by death or by predator before having the possibility of reproduction (see [24,30]). However, the infections still contribute to growth of susceptible prey N according to the logistic growth law. (H4): The disease spreads only among the prey population and disease is not genetically inherited. The infected individuals do not recover or become immune. The new infections arise from the contact of infected and susceptible individuals and follows the mass action infection mechanism, which leads to the following SI epidemic model: where β is the disease transmission rate and c is death rate of infections. We further assume that both β and c are temporally-spatially heterogeneous. (H5): Assume that the predator individuals only feed on the infected prey with general functional response function f (t, x, I, W ) at time t and location x. We also assume that the natural death rate of predator population is d(t, x) which depends on time and space variables.
Biologically, (H5) means that the predator only catches the infected prey, this is because the infected individuals are less active or the behavior of the prey individual is modified such that they live in parts of the habitat which are accessible to the predator (see [19]). The goal of the current work is devoted to the dynamics of model (1) with no flux boundary condition, namely, (2) Here functions a(t, x) and β(t, x) are Hölder continuous and nonnegative nontrivial on R×Ω, and periodic in time with the same period T > 0; the function b(t, x), c(t, x) and d(t, x) are Hölder continuous and positive on R × Ω, and periodic in time with the same period T > 0; the diffusion coefficients D S (t, x), D I (t, x) and D W (t, x) belong to C θ/2,θ (R × Ω, R + ) for some θ ∈ (0, 1) and are periodic in time with the same period T > 0. Moreover, we assume satisfies the following assumptions (see, e.g., [23]) (F1): f i (·, ·, 0, ·) ≡ 0 and f i (·, ·, ·, 0) ≡ 0, i = 1, 2.
(F2): f i (· + T, ·, ·, ·) ≡ f i (·, ·, ·, ·), i = 1, 2. When a(t, x), b(t, x), c(t, x), d(t, x), β(t, x) and D i (t, x)(i = S, I, W ) are positive functions, f 1 and f 2 are independent of time and space variables and chosen as ratio-dependent Michaelis-Menten functional response function m4IW αW +I , the system (2) is reduced to where m 2 and m 4 are positive constants, and ∂ n is the outward directional derivative normal to ∂Ω. Li and Gao [12] showed a number of existence and nonexistence results about the non-constant steady states of system (3). They also studied Diffusion-Driven Instability and bifurcation of non-constant solutions. If we further assume that the diffusion rates are identical in system (3), Zhang et al. [33] achieved the sufficient conditions for the permanence, the global stability of boundary equilibria and the positive equilibrium, and the local stability of the positive equilibrium. In fact, if we ignore the diffusion factor, the model system (1) degenerates into an ODE system, which is formulated and analyzed by Xiao and Chen [30]. It is interesting that in [30], a periodic solution could occur whether the system is permanent or not. The rest of this paper is organized as follows. In the next section, we show the well-posedness of system (2), that is, the global existence, uniqueness and positivity of solutions of (2), and we further prove the existence of global attractor. In Section 3, under the case where the predator is absence, we study the time-periodic and reaction-diffusion SI model, specifically, we introduce the basic reproduction number R 0 via a next generation operator for the model and establish the threshold dynamics of the model system in terms of R 0 . In Section 4, we obtain the conditions for the persistence and extinction of system (2).
Lemma 2.1. The following equation admits a positive T -periodic solution S * if and only if µ 0 < 0. In the affirmative case S * is unique and global attractive for (6) with continuous non-negative non-trivial initial data. Moreover, if µ 0 ≥ 0, the trivial solution 0 is globally attractive.
Let U (t, s), V (t, s) and W (t, s) be the evolution operators determined by the following reaction-diffusion equations for t ≥ 0. Then (2) becomes where A 2 (t) and A 3 (t), respectively, are defined by . Rewrite the abstract equation (7) as an integral equation We regard the solution of (8) as a mild solution of (7). Clearly, F is locally Lipschitz continuous. It is not difficult to verify that F is quasi-positive (see, e.g., [ , it then follows that for any φ ∈ Y + , system (2) admits a unique non-continuable mild solution satisfying u 0 = φ and u(t, φ) ∈ Y + for any t in its maximum interval of existence [0, σ φ ). Moreover, by the analyticity of Z(t, s), s, t ∈ R, s < t, u(t, x; φ) is a classical solution when t > 0.
Recall that a family of operators {Φ t } t≥0 is an ω-periodic semiflow on a metric space (G, ρ) with the metric ρ, provided that {Φ t } t≥0 satisfies: We are in position to show that the solutions of system (2) exist globally on [0, ∞).
Proof. In view of equation (6) and Lemma 2.1, it follows that either the zero solution of (6) or the unique positive T -periodic solution S * of (6) in X + is globally attractive. Since the first equation in (2) is dominated by (6), the comparison principle implies that S(t, ·; φ) is bounded on [0, σ φ ). Since f 1 and f 2 satisfies (F2)-(F3), we see that the equations for I and W are dominated by a scalar linear reactiondiffusion equation, respectively. It then follows that σ φ = ∞ for each φ ∈ Y + . Thus, there exists a constant M 1 > 0 such that for any φ ∈ Y + , we can find a positive integer l 1 = l 1 (φ) For any φ ∈ Y + , let (S(t, x), I(t, x), W (t, x)) := (S(t, φ)(x), An integration of the first equation and Green's formula imply By using the above inequality (9) and Green's formula, integrating the equation for which is equivalent to d S (t) +Ī(t) dt ≤ −c S (t) +Ī(t) + (â +c)S(t), ∀t > 0.
3. The SI model. When the predator population is absent, the system (2) become the following periodic and reaction-diffusion SI epidemic model: Define Z := C(Ω, R 2 ) and Z + := C(Ω, R 2 + ). Then (Z, Z + ) is a strongly ordered Banach space with supermum norm · Z . In view of Theorem 2.2, it is easy to show the conclusion on the well-posedness of system (11).
In the following, motivated by the idea in [27] (see also [18,32]), we introduce the basic reproduction number R 0 for the model (11). Let C T (R, X) be the ordered Banach space consisting of all T -periodic and continuous functions from R to X, where ψ C T (R,X) := max θ∈[0,T ] ψ X for any ψ ∈ C T (R, X). Define C + T (R, X) := {ψ ∈ C T (R, X) : ψ(t)(x) ≥ 0, ∀t ∈ R, x ∈ Ω}. Letting I = 0 in (11), we obtain the equation for S(t, x) as follows: By means of Lemma 2.1, it follows that (12) admits a unique positive T -periodic solution S * (t, x) and it attracts the solutions of (12) with continuous non-negative non-trivial initial data if the principal eigenvalue µ 0 of (5) satisfies (P): µ 0 < 0. Then we have the following observation.
Linearizing system (11) at (S * , 0), we obtain the following equation for the infection component: From the discussion in Section 2, it follows that there exist positive constants H ≥ 1 and ω ∈ R such that V (t, s) ≤ He ω(t−s) , ∀t ≥ s, t, s ∈ R.
Consider the following periodic-parabolic eigenvalue problem Referring to [9], we know that (14) has a unique principal eigenvalue λ 0 (S * ) with positive periodic eigenfunction.
Suppose that ϕ(s, x) = ϕ(s)(x) ∈ C T (R, X) is the initial distribution of infectious individuals at time s ∈ R and spatial location x ∈ Ω. Then the term (F (s)ϕ(s)) (x) = β(s, x)S * (s, x)ϕ(s, x) denotes the density distribution of the new infections produced by the infected individuals who were introduced at time s. Thus, for given t ≥ s, (V (t, s)F (s) ϕ(s)) (x) represents the density distribution at location x of those infected individuals who were newly infected at time s and remain infected at time t. Consequently, the term means the density distribution of the accumulative new infectious at time t and location x produced by all those infected individuals ϕ(s, x) introduced at all the previous time to t. As a result, we can define the next generation operator L as Based on the assumptions on β and c, it is not difficult to see that L is a positive and bounded linear operator on C T (R, X). We define the spectral radius of L to be the basic reproduction number for the model (11), namely, As a straight consequence of [22,Theorem 5.7] (see also [18,Lemma 2.2]), we have the following lemma. Similar to [18], in order to obtain an equivalent characterization of R 0 , we first consider a periodic-parabolic eigenvalue problem: According to [9,Theorem 16.3], we see that (15) admits a unique and positive principal eigenvalue ρ 0 associated with an eigenfunction ψ 0 ∈ C T (R, X) and ψ 0 > 0 on R × Ω. By similar arguments to [18, Lemma 2.1], we show the conclusion as follows.
We are in position to establish the threshold dynamics of (11) in terms of R 0 .
(ii). In the case where R 0 > 1, Lemma 3.3 implies that λ 0 (S * ) < 0. Let It is easy to see that Ψ(nT )W 0 ⊂ W 0 , ∀n ∈ N. Furthermore, we have the following three claims.
arguments as above can lead to lim t→∞ W (t, x) = 0 uniformly for x ∈ Ω. This proof is completed.