PATTERN FORMATION OF A DIFFUSIVE ECO-EPIDEMIOLOGICAL MODEL WITH PREDATOR-PREY INTERACTION

. We consider a predator-prey system with a ratio-dependent functional response when a prey population is infected. First, we examine the glo- bal attractor and persistence properties of the time-dependent system. The existence of nonconstant positive steady-states are studied under Neumann boundary conditions in terms of the diﬀusion eﬀect; namely, pattern formati-ons, arising from diﬀusion-driven instability, are investigated. A comparison principle for the parabolic problem and the Leray-Schauder index theory are employed for analysis.


1.
Introduction. Over the last three decades, predator-prey models have been studied extensively by many researchers. In particular, ratio-dependent predatorprey models, in which the per capita predator growth rate depends on a function of the ratio of prey to predator abundance, have been proposed by Arditi and Ginzburg [1]. Ratio-dependent models have been mathematically studied for both the spatially homogeneous case [13,14,15] and spatially inhomogeneous case [6,20,24]. The actual evidence and justification of these models have also been studied [2,3,9,11].
On the other hand, because of Kermack-McKendrick's model, epidemic models have also received a lot of attention. We are interested in the study of ecological systems with the influence of epidemiological factors. A reasonable number of studies have been done on the spatially homogeneous case [5,7,8,12,26,27].

WONLYUL KO AND INKYUNG AHN
Arino et. al. [4] suggested the following non-dimensionalized model, which is a non-spatial version : − λSI, S(0) > 0 dI dt = λSI − α I IP k 1 P + S + I − δ 1 I, I(0) > 0 dP dt = k(α S S + α I I)P k 1 P + S + I − δ 2 P, P (0) > 0, where S, I, and P denote the population density of the susceptible prey, infected prey, and predator, respectively. The initial conditions are R 3 + = {(S, I, P ) ∈ R 3 : S ≥ 0, I ≥ 0, ≥ 0}. The authors studied the behavior of the system near the equilibria, and obtained the criteria of the persistence of the system.
In this paper, with this motivation, we consider a diffusive eco-epidemiological model with infection in the prey and ratio-dependent functional responses; where Ω ⊆ R N is a bounded region with smooth boundary ∂Ω, and r, m, K, b, d i , D, c, α, and β are positive constants. The initial functions u 0 , v 0 , and w 0 are not identically zero in Ω; u, v, and w represent the densities of the susceptible prey, infected prey, and predator, respectively, and η is the outward directional derivative normal to ∂Ω. Furthermore, α and β are the searching efficiency constants of the predation rate for the susceptible and infective prey, respectively. α m and β m are the maximum per capita capturing rates of the predator for the susceptible prey and infected prey, respectively. m is the predation rate for the susceptible prey and infected prey. Finally, b is the force of infection, d 1 and d 2 are the death rates of the infected prey and predator, respectively, and c is a conversion rate. The homogeneous Neumann boundary condition describes an environment with no flux at the boundary of the region.
Model (1) is based on the following assumptions: • In the absence of disease, the prey population grows according to logistic law with carrying capacity K > 0 and an intrinsic growth rate r > 0. • In the presence of disease, the prey consists of two classes : susceptible prey and infected prey. • Only susceptible prey can reproduce themselves logistically and contribute to its carrying capacity. Infected prey do not grow, recover, or reproduce. • Disease can only be spread among the prey, and it is not inherited. Disease transmission follows the simple law of mass-action.
For additional background information pertaining to (1), we refer to [4] and the references therein.
The goal of this study is to determine the coexistence states by investigating the non-constant positive solutions of the following time-independent system : Note that the given growth rates in (2) are not defined at (u, v, w) = (0, 0, 0). However, since the domain of uw mw+u+v and vw mw+u+v may be extended to {(u, v, w) : u ≥ 0, v ≥ 0, w ≥ 0} so that (0, 0, 0) becomes a trivial solution to (2) [15]. Furthermore, note that the growth rates of the given system are not quasimonotone nondecreasing.
The remainder of this paper is organized as follows. In Section 2, the global attractor and persistence properties are given for solutions to the time-dependent system in (1). In Section 3, we estimate the bounds of positive solutions to system (2). Section 4 provides the existence and non-existence of non-constant positive solutions to (2). Finally, the results obtained are analyzed briefly in terms of biological interpretations in Section 5.
2. Global attractor and persistence properties of time-dependent system. In this section, the global attractor and persistence properties are investigated for solutions to the time-dependent system in (1).
For convenience, we denote the growth rate terms as follows: Using the uniform bound of u, v and w, one can show that (uf 1 , vf 2 , wf 3 ) satisfies the Lipschitz condition. Using the upper and lower solution method in [22], it can also be shown that (1) has a nonnegative solution.
The next theorem states that the solution to (1) is uniformly bounded.
Theorem 2.1. The solution (u, v, w) of (1) is uniformly bounded; specifically, where B i is defined by Proof. First, note that from the strong maximum principle and the Hopf Boundary Lemma for parabolic problems, u(t, x), v(t, x), and w(t, x) are nonnegative. Next, we show that the solution to (1) is bounded above on [0, ∞) × Ω. Since , by a comparison argument for the parabolic equation, 0 ≤ u(t, x) ≤ B 1 holds .
Adding the first equation to the second equation in (1) yields, The last inequality follows from the fact that the maximum value of u(r Then by the maximum principle, Z ≤ B 2 . Since v ≤ Z, the desired result is obtained. Finally, the upper bound B 3 of w follows from the maximum principle, the fact that u ≤ Z ≤ B 2 , and the following inequality: Next, we examine the dissipation and persistence of parabolic system (1).
Proof. First, lim sup t→∞ u(t, x) ≤ K follows from a comparison argument for parabolic problems, since . To obtain the inequality involving v, we use the same argument used in the proof of Theorem 2.1. Consider the following parabolic problem: is the solution to this . The same inequality also holds for u. Thus, there exists a T such that Finally, using the same method used in Theorem 2.1 to find the uniform bound of Therefore, since ε is arbitrary, the desired result is achieved.
The following theorem states the persistence property of system (1).
Proof. Note from Theorem 2.2 that for an arbitrary ε > 0, there exists In light of this fact, it follows that Thus, there exists T 2 ≥ T 1 such that u ≥ Θ 1 − ε on Ω for t ≥ T 2 . Next, find Θ 3 , where Θ 3 is the lower bound of w. Note that since β > α, the term cαu mw+u+v + cβv mw+u+v in f 3 is monotone increasing for v ≥ 0. Since × Ω, consider the following parabolic system: Since the given growth rates are not quasimonotone nondecreasing, we are not able to directly apply the comparison theorem for a reaction diffusion system. Thus, 380 WONLYUL KO AND INKYUNG AHN we extend system (3) to four equations (see [23]) : for a sufficiently small positive constant δ 1; by the comparison principle, P ≤ P and Q ≤ Q, where ( P , Q) satisfies the following parabolic system: The equilibrium point ( P * , is globally asymptotically stable. This can be verified by considering the Lyapunov function 3. Bound estimates of positive steady-states. First, we estimate the bounds of the positive solutions to the elliptic system (2). To estimate the a-priori bounds for positive solutions to (2), the Maximum Principle [16] and Harnack inequality, introduced by Lin et al. [17], are used. Note that a positive solution to (2) is contained in [C 2 (Ω)] 3 by the standard regularity theorem for elliptic equations [10,25]; hence the Harnack inequality can be applied to (2).
Theorem 3.1. If c min{α, β} > d 2 , then positive solutions (u, v, w) of (2) satisfy Proof. Applying the Maximum Principle to the first equation in (2) yields the first assertion.
To establish the second assertion, we add the first and second equations in (2), and then we derive in Ω. Thus from the Maximum principle, we obtain on Ω, which gives the second assertion. Finally, using the maximum principle again in the third equation in (2), we have the third assertion.
We now estimate a positive lower bound for classical positive solutions to (2). For simplicity, we let Γ denote (r, K, α, β, b, m, d i ). if Proof. It is easy to verify that f1 d , f2 d , f3 D ∈ C(Ω) for d ≥ d * , D ≥ d * . By the Harnack inequality, there exists a positive constant C * (N, Ω, d * , Γ) such that are satisfied. Suppose to the contrary that (7) is not satisfied. Then by (9), there exist sequences {d n , D n } and a corresponding positive solution (u n , v n , w n ) to (2) such that d n ≥ d * and D n ∈ [d * , D * ] for n ∈ N; furthermore, max Ω u n → 0, max Ω v n → 0, or max Ω w n → 0 as n → ∞.
By the regularity theory of elliptic equations, there exists a subsequence of {(u n , v n , w n )} and nonnegative functions u, v, w ∈ C 2 (Ω) such that (u n , v n , w n ) → ( u, v, w) as n → ∞. Moreover, we assume that (d n , as n → ∞. Since max Ω u n → 0, max Ω v n → 0, or max Ω w → 0 as n → ∞, u ≡ 0, v ≡ 0, or w ≡ 0. If we only consider the cases that satisfy the last sentence, we are reduced to the following:

WONLYUL KO AND INKYUNG AHN
The following integral equations are obtained by integrating by parts: (i): Note that v n , w n > 0 hold by (9), and u, v, and w satisfy (9) since u n , v n , and w n satisfy (9). Thus, v > 0, since v ≡ 0. Furthermore, since v n → v > 0 and u n → 0 as n → ∞, v n f 2 (u n , v n , w n ) < 0 for a sufficiently large n. This is a contradiction to the second equation in (10).
(ii): Similarly, for a sufficiently large n, w n f 3 (u n , v n , w n ) < 0 since w n > 0, v n → v ≡ 0, and u n → u ≡ 0. This is a contradiction to the third equation in (10).
If necessary, we consider a subsequence and assume Consider the case when d < ∞. If W ≡ 0, then U + V > 0 on Ω; moreover, if U ≡ 0, V > 0 holds on Ω. It is also true that Ω V (b u − d 1 ) = Ω −d 1 V < 0; however, this contradicts the second equation in (12). Conversely, if V ≡ 0, then U > 0 holds on Ω and Ω U (r − r K u − b v) = Ω r U > 0. Again this contradicts the first equation in (12). Thus, W > 0 on Ω.
Therefore, bu n −d 1 − βWn mWn+Un+Vn < 0 uniformly on Ω as n → ∞. This contradicts the second integral equation of (12). Now consider the case when d = ∞. In this case, U = A and V = B, where A and B are nonnegative constants. Similar to the case when d < ∞, one can show that W > 0 on Ω. However, this violates the second integral equation in (12), since u n → u ≡ 0 as n → ∞.

Nonconstant positive solutions.
In this section, we obtain the results for the nonexistence and existence of a nonconstant positive solution to system (2).
First, we observe that if the following conditions are satisfied : where then there exists a unique positive equilibrium point u * = (u * , v * , w * ), where

4.1.
Nonexistence of nonconstant positive solutions. In this subsection, we investigate a condition that implies the nonexistence of nonconstant positive solutions to (2). This result is also used in the index approach to show the existence of nontrivial positive steady states, which is the main result of this section. Before developing our argument, we define the following notation, which is similar to the notation defined in [18,21]. (iv) X ij = {c · ϕ ij |c ∈ R 3 } (v) X = u = (u, v, w) ∈ [C 1 (Ω)] 3 | ∂u ∂η = ∂v ∂η = ∂w ∂η = 0 on ∂Ω .
The integral equation in (15) is less than or equal to the following: where V is the upper bound of v, defined in Theorem 3.2, and ε is an arbitrary positive constant. The last inequality in (16) follows from the ε-Young's Inequality. By the Poincaré inequality, the following is satisfied:

WONLYUL KO AND INKYUNG AHN
Then system (17) can be written as the following elliptic system : Thus, (19) is equivalent to finding the positive roots of the characteristic equation for k ≥ 1. Hence, the above Leray-Schauder Theorem can be rewritten as follows : where n λ k = m λ k dim E(µ k ), m λ k is the multiplicity of λ k as a positive root of B k (λ) = 0, and E(µ k ) is defined in Notation 4.1. From Theorem 4.1, we know that there does not exist a nonconstant positive solution to (2) if d > d for a sufficiently large d when Dµ 2 > 2cα + 2cβ − d 2 . Thus, for the index approach, we investigate the index value at u * when d is a sufficiently large. With some careful calculation, we can show that the given coefficients were all positive. Thus, B 1 (λ) > 0 for all λ ≥ 0. Now, consider the case when k ≥ 2, i.e., µ k > 0. The polynomial B k (λ) can be written as follows: Thus, there exists a large positive constant d depending on Γ, N , Ω, and D such that B k (λ) > 0 for all d ≥ d and λ ≥ 0. Therefore, B k (λ) > 0 for all λ ≥ 0, k ≥ 1 and d ≥ d, implying γ = k≥1 λ k >0 n λ k = 0. 5. Concluding remarks. A diffusive predator-prey model with a ratio-dependent functional response and infected prey population was investigated under homogeneous Neumann boundary conditions. First, the global attractor and persistence properties of the given time-dependent system were examined. We showed that there is no nonconstant positive solution (i.e., no pattern formation occurs) under a suitable diffusion condition. In particular, if the diffusion of the predator is not too small, no pattern occurs when the diffusion of the susceptible and infected prey is sufficiently large. On the other hand, when there is no nonconstant positive solution, a Turing pattern is induced by the predator's large diffusion value. Therefore, diffusion of the predator creates a spatially nonconstant positive solution from Turing instabilities under suitable conditions.