SPATIOTEMPORAL DYNAMICS OF A DIFFUSIVE PREDATOR-PREY MODEL WITH GENERALIST PREDATOR

. In this paper, we study the spatiotemporal dynamics of a diﬀusive predator-prey model with generalist predator subject to homogeneous Neu- mann boundary condition. Some basic dynamics including the dissipation, persistence and non-persistence(i.e., one species goes extinct), the local and global stability of non-negative constant steady states of the model are investi-gated. The conditions of Turing instability due to diﬀusion at positive constant steady states are presented. A critical value ρ of the ratio d 2 d 1 of diﬀusions of predator to prey is obtained, such that if d 2 d 1 > ρ , then along with other suitable conditions Turing bifurcation will emerge at a positive steady state, in particu- lar so it is with the large diﬀusion rate of predator or the small diﬀusion rate of prey; while if d 2 d 1 < ρ , both the reaction-diﬀusion system and its corresponding ODE system are stable at the positive steady state. In addition, we provide some results on the existence and non-existence of positive non-constant steady states. These existence results indicate that the occurrence of Turing bifurca- tion, along with other suitable conditions, implies the existence of non-constant positive steady states bifurcating from the constant solution. At last, by nu- merical simulations, we demonstrate Turing pattern formation on the eﬀect of the varied diﬀusive ratio d 2 d 1 . As d 2 d 1 increases, Turing patterns change from spots pattern, stripes pattern into spots-stripes pattern. It indicates that the pattern formation of the model is rich and complex.


Introduction. Recently, Kang and Fewell
studied a host-parasite coevolutionary model, in which the relationship between host(prey) and parasite(predator) is described by the following ecological model with Holling Type II functional response    du dt = u r 1 where u and v represent the population densities of prey and predator at time t, respectively, r i , K i (i = 1, 2), a, e, h, d are positive constants with specific ecological interpretations : r 1 and r 2 denote the intrinsic growth rate of species u and v, respectively, in the absence of other species; K 1 and K 2 are the carrying capacities of species u and v individuals, respectively; d stands for the dead rate of predator individuals due to hunting or attacking all potential prey resources; a is the capturing efficiency of a predator and h is the predator handling time. In [18], the ecological dynamics for model (1), including the boundedness, permanence, stability of boundary and interior equilibria, and extinction of one species, were performed.
In system (1), both species u and v follow Logistic growth in the absence of other species, which indicates that species v has alternative food resources and could persist in the absence of prey u, i.e., predator species v is generalist. In natural, many predators are in fact generalist, and their prey consists of different species(see, for example, [30,34,33,2]). In the last couple of decades, the study on prey-predator work involving generalist predators has attracted more and more attentions, and provided additional biological insights on dynamical outcomes for models with predator being generalist versus specialist. For example, the study of Symondson et al [35] shows that generalist predators could be effective control agents and have some unique biocontrol functions that are denied to specialists. The work of Kang et.al. [18,19] indicates that models with generalist predator could exhibit more complicate dynamics and more likely to have "top down" regulation by comparing to the similar models with specialist predator. We also refer to [14,13,31,9,24,29,6] for some references on the study of predator-prey models with generalist predators.
The interactions between predator and prey generally occur over a wide range of spatial and temporal scales and the spatial diffusions of generalist predator and its preys play important roles in shaping ecological communities. The parabolic and elliptic predator-prey models with generalist predators also have been widely studied. Blat and Brown [1] studied the non-negative steady-state solutions of a reaction-diffusion system with generalist predator under Dirichlet boundary condition. Treating prey and predator birth-rates as bifurcation parameters in [1], the ranges of parameters were given for which there exist non-trivial steady-state solutions. In spatially heterogeneous environment, the reaction-diffusion systems with generalist predators were studied in [7,28], and some results on the existence, non-existence and multiplicity of positive steady states and global bifurcation were obtained. In [8,15], the reaction-diffusion models with generalist predators and protection zones for preys were studied. We also refer to [17,36] for the study on the positive steady-state solutions of coupled reaction-diffusion Lotka-Volterra systems, and [23] for the study of the travelling wave solutions of a reaction-diffusion system with generalist predator. In this paper, we study the spatiotemporal dynamics of the reaction-diffusion predator-prey model corresponding to ODE model (1) x ∈ Ω, t > 0, ∂u ∂ν = ∂v ∂ν = 0, x ∈ ∂Ω, t > 0, where, Ω is a bounded domain of R N (N ≥ 1) with smooth boundary ∂Ω, ∂ ∂ν is the outward directional derivative normal to ∂Ω, u 0 , v 0 ∈ C(Ω) stand for the initial conditions, d 1 and d 2 are positive constants and stand for the random diffusive rates of the prey and predator, respectively.
When r 2 > d and Ω is a two-dimensional bounded connected square domain, by changing r 2 − d and r2 K2 into r 2 and m, respectively, (2) becomes the model studied by Chakraborty [5]. In [5], the condition of Turing bifurcation at positive constant steady state due to diffusion was given, and different pattern formations and spatiotemporal chaos were presented by numerical simulation. In this paper, we discuss the dissipation, persistence and non-persistence(i.e., one species goes extinct), the local and global stability of non-negative constant steady states of (2). The conditions of Turing bifurcation at positive constant steady state of (2) are presented. A critical value ρ of d2 d1 is obtained such that if d2 d1 > ρ, then along with other suitable conditions Turing bifurcation emerges at a positive steady state, in particular so it is with the large diffusion rate d 2 of predator or the small diffusion rate d 1 of prey; while if d2 d1 < ρ, (2) has the same stability to ODE system (1). We refer reader to [12,37,4,38,32,20] for some references on the recent study of Turing bifurcation and spatiotemporal pattern formation of some ecological models.
In order to study the stationary pattern induced by diffusions, we also consider the existence and non-existence of positive non-constant solutions of the steady state of (2), i.e., the following semi-linear elliptic system In [27], the non-existence of non-constant positive steady state solutions of the following reaction-diffusion predator-prey model with Holling type-II functional response and generalist predator was studied where the constant β may be non-positive. It was proved under N ≤ 3 in [27] that for any given d 1 , d 2 , α, β, Ω, there exists a positive constant M , which depends on d 1 , d 2 , α, β, Ω, such that if m ≥ M , then (4) has no non-constant positive solution when β ≤ 0 and has no positive solution when β > 0. For model (3), we are interested in the effect of diffusion rates d 1 and d 2 on the stationary pattern. It is proved that if r 2 < d < r 2 + eaK1 1+haK1 and d 2 > 1 µ1 r 2 − d + eaK1 1+ahK1 , where µ 1 is the smallest positive eigenvalue of −∆ on Ω with zero-flux boundary conditions, (3) has no non-constant positive steady state solution for sufficiently large d 1 . In addition, some existence results of at least one non-constant positive steady state solution of (3) are obtained by the index formula given by Pang and Wang [26] and Leray-Schauder topological degree theory. These existence results indicate that the occurrence of Turing instability at a positive constant steady state of reactiondiffusion system, along with other suitable conditions, implies the existence of nonconstant positive steady state bifurcating from the constant solution.
The remaining sections of this paper is organized as follows. In Section 2, we show the results on the dissipation, permanence, non-persistence, and the local and global stability of non-negative constant steady states of model (2), as well as Turing instability at positive constant solutions of (2). In Section 3, we first give a priori upper and lower bounds for the positive solutions of (3) in order to calculate the topological degree, then present the non-existence and existence of positive nonconstant solutions of (3). In Section 4, we perform a series of numerical simulation to show the occurrence of Turing patterns caused by diffusions. It is found that the model dynamics exhibit spatiotemporal Turing complexity of pattern formation, including spots, strips and spots-stripes Turing patterns. At last, we end the paper with a brief discussion in Section 5.
2. Permanence, stability, Turing instability. In this section, we study the dissipation, persistence and non-persistence, the local and global stability of nonnegative constant steady states of model (2). Also, Turing instability at the positive constant solution is studied.

2.1.
Permanence. First, we show the dissipation and permanence of (2). Define , we can directly get the first assertion from the comparison argument for parabolic problems [11]. Thus, there exists T > 0 such that u(x, t) ≤ K 1 + for (x, t) ∈ Ω × [T, ∞). It follows that v(x, t) satisfies It follows from the comparison argument [11] that v(x, t) ≤ z(t), and hence lim sup by the arbitrariness of . If d > r 2 + eaK1 1+haK1 , we have the differential inequality ∂v ∂t − d 2 ∆v ≤ − r 2 v K 2 , and the same argument above yields lim sup t→∞ max x∈Ω v(x, t) ≤ 0. In either case, the second assertion holds. This completes the proof.
The following result gives the sufficient conditions for the permanence of (2).
Proof. We only show the proof for the case r 2 ≤ d. Choose any : 0 < < m u sufficiently small, such that r 1 − a(M v + ) > 0 and Let w(t) be the unique positive solution of the following problem Then w(t) is a lower solution of (5) and lim t→∞ w(t) = K1 r1 (r 1 − aM v ) = m u by the arbitrariness of . It follows from the comparison principle [11] that the desired first assertion holds. Hence, there exists T 0 > T such that u(x, t) ≥ m u − in Ω × [T 0 , ∞). Thus, we have that v(x, t) is an upper solution of Again, using the comparison principle [11], we can get the second assertion. The proof is completed.

2.2.
Non-persistence. It is known from [18] that system (2) has two semi-trivial constant steady states E 10 = (K 1 , 0) and In this subsection, we discuss the global stability of the constant steady states E 10 and E 01 , which implies that one species goes extinct and system (2) has no positive non-constant steady state regardless of the diffusion coefficients.
Proof. 1. From the proof of Theorem 2.1, there exists a positive function z(t) satisfying lim t→∞ z(t) = 0 such that max Applying the comparison principle [11], we have that The arbitrariness of then implies that lim inf t→∞ min x∈Ω u(x, t) ≥ K 1 . This, along with the result of Theorem 2.1, implies that u → K 1 uniformly onΩ as t → ∞. 2. By the second equation of (2), we have Since r 2 > d, it follows from a comparison argument that This implies that for any , An application of the comparison principle gives max x∈Ω u(x, t) ≤ w(t).
If ahK 1 > 1 and r1(1+ahK1) 2 <m v , then which also implies that lim t→∞ w(t) = 0. Thus, u → 0 uniformly onΩ as t → ∞. It Again by a comparison argument, we have The arbitrariness of then implies that lim sup The proof is completed.

Local stability and Turing bifurcation.
Except the two semi-trivial constant steady states E 10 = (K 1 , 0) and E 01 = 0, (1 − d r2 )K 2 (r 2 > d), system (2) has a trivial constant steady state E 00 = (0, 0), and at most three positive constant steady states E * = (u * , v * )(see [18]). The interior equilibrium E * (u * , v * ) of system (1) satisfies the following two equations In this subsection, we analyze the local stability of these constant steady states. Also, Turing instability of the positive constant steady state E * = (u * , v * ) is studied.
Assume that 0 = µ 0 < µ 1 < · · · are the eigenvalues of the operator −∆ on Ω with the homogeneous Neumann boundary condition. Set S p = {µ 0 , µ 1 , µ 2 , · · · } and E(µ i ) the eigenspace corresponding to µ i in C 1 (Ω). Let For the sake of simplicity, we rewrite system (2) in a compact form   and For convenience, set and Denote the discriminant of H(d 1 , d 2 , µ) = 0 as follows If Q(d 1 , d 2 ) > 0, then H(d 1 , d 2 , µ) = 0 has two real roots, denoted by Define Now, we give the main results on the local stability of the non-negative constant steady states of system (2). For this, we make the following assumptions We also take the following notations Theorem 2.4. (Local stability and Turing instability) 1. For the positive constant steady state E * = (u * , v * ) of system (2), (i) if K 1 ≤ 2u * + 1 ah , it is locally uniformly asymptotically stable; (ii) under Condition (H), it is locally uniformly asymptotically stable if (iii) under Condition (H), it is locally uniformly asymptotically stable if (iv) under Condition (H) and it is locally uniformly asymptotically stable if (2) is locally uniformly asymptotically stable if r1 aK2 < 1 − d r2 . 4. The trivial constant steady state E 00 = (0, 0) of system (2) is always unstable.
Note that for each i ≥ 0, X i is invariant under the operator L. λ is an eigenvalue of L if and only if λ is an eigenvalue of the matrix where ah , it is easy to see that Tr(A i ) < 0, Det(A i ) > 0 for all i ≥ 0. Therefore, for each i ≥ 0, the two roots λ i 1 and λ i 2 of ψ i (λ) = 0 both have negative real parts. By a standard argument(see, for example, [10,3]), one can prove that there exists an η > 0 such that Therefore, the spectrum of L, which consists of eigenvalues, lies in { Reλ ≤ −η}.
In the following proofs for (ii)-(iv), we assume that Condition (H) holds. Then (ii) By (14), one can check that d 1 j 22 + d 2 j 11 < 0 and Tr(J| E * ) < 0. It follows that Tr(A i ) < 0, Det(A i ) > 0 for all i ≥ 0. Therefore, for each i ≥ 0, the two roots λ i 1 and λ i 2 of ψ i (λ) = 0 both have negative real parts. Similar to the arguments of case (i), E * = (u * , v * ) is uniformly asymptotically stable.
(iv) From (16), it is easy to verify that So, for each i ≥ 0, the two roots λ i 1 and λ i 2 of ψ i (λ) = 0 both have negative real parts and E * = (u * , v * ) is uniformly asymptotically stable. If which implies that Det(A i0 ) < 0. Thus, the equation ψ i0 (λ) = 0 has a positive real root. On the other hand, according to the arguments above, one can easily obtain that ODE system (1) is locally asymptotically stable at the positive equilibrium E * = (u * , v * ) if Condition (H) holds and 0 < d < r 2 + N 1 (u * ). Therefore, the positive constant steady state E * = (u * , v * ) of system (2) is Turing instability. 2. Consider the semi-trivial constant steady state E 10 = (K 1 , 0). Clearly, Since d > r 2 + eaK1 1+ahK1 , we have Tr(J| E10 ) < 0 and Det(J| E10 ) > 0. The remaining arguments are rather similar as above. Therefore, E 10 is uniformly asymptotically stable.
4. Consider the trivial constant steady state E 00 = (0, 0). It is easy to see that Since J| E00 has a positive eigenvalue r 1 , E 00 = (0, 0) is unstable. The proof is completed.
Notation. Theorem 2.4 and its proof indicates the follows. 1. The reaction-diffusion system (2) has the same local stabilities at E 00 , E 10 and E 01 with ODE system (1)(see Kang and Fewell [18]). Therefore, the diffusions of prey and predator don't influence the stability of E 00 , E 10 and E 01 ; 2. If d 2 ≤ d 1 , then N 2 (u * ) ≥ N 1 (u * ), and (15) and (16) fail while (14) holds. This implies that the reaction-diffusion system (2) has the same local stability at E * with ODE system (1), and the spatial diffusions of species can't produce Turing bifurcation. However, if d 2 > d 1 then the diffusions of species may drive Turing instability to occur when (16) and The sufficient condition of Turing instability given in Theorem 2.4 doesn't explicitly show the effect of diffusion rates on the occurrence of Turing bifurcation. In the following result, a critical value ρ of d2 d1 is obtained such that if d2 d1 > ρ, then along with other suitable conditions Turing bifurcation emerges at a positive steady state; while if d2 d1 < ρ, (2) has the same stability to ODE system (1). Comparing to Theorem 2.4, the conditions given below is more easy to be checked.
The following corollary is a direct application of Theorem 2.5. It indicates that the large diffusion rate d 2 of predator or the small diffusion rate d 1 of prey will lead to the occurrence of Turing instability at the positive constant steady state E * = (u * , v * ) of (2). Corollary 1. Assume that (H) holds and 0 < d < r 2 + N 1 (u * ). 1. There exists d * 2 > 0 such that for d 2 > d * 2 , the positive constant steady state E * = (u * , v * ) of (2) is locally uniformly asymptotically stable if j11 d1 < µ 1 ; while it is Turing instability if j11 d1 > µ 1 . 2. There exists d * 1 > 0 such that for d 1 < d * 1 , the positive constant steady state E * = (u * , v * ) of (2) is Turing instability.

2.4.
Global stability at positive constant steady states. In this subsection, we establish the global asymptotic stability of the positive constant steady state of system (2), which implies the non-existence of non-constant steady state of (2) regardless of the diffusion coefficients.
3. Non-constant positive steady-states. In order to study the stationary pattern of (2) induced by diffusions, in this section we discuss the existence and nonexistence of non-constant positive solutions of system (3).
3.1. Bounds for positive steady state. In this subsection we give a priori upper and lower bounds for the positive solutions of (3). To this aim, we recall the following maximum principle [22] and Harnack Inequality [21].  [22]) Let Ω be a bounded Lipschitz domain in R N and g ∈ C(Ω × R).
Then, again by Lemma 3.1, we have The first assertion is proved. Let Clearly, Then ||c 1 (x)|| ∞ ≤C, ||c 2 (x)|| ∞ ≤C, whereC depends on N, Ω,d, Λ. Thus, in view of Lemma 3.2, there exists a positive constant C * = C * (N, Ω,d, Λ) such that the Harnack inequality holds provided d 1 , d 2 ≥d. Now, we verify the lower bounds of u(x) and v(x). On the contrary, suppose that the conclusion is not true, then there exist sequences {d 1i } and {d 2i } with d 1i , d 2i ≥ d and the positive solution by (19).

3.2.
Non-existence of non-constant positive steady states. From Theorem 2.3, we know that if either d < r 2 along with other suitable conditions or d > r 2 + eaK1 1+haK1 , (3) has no non-constant positive solution regardless of diffusion coefficients. In this subsection, we consider the case r 2 < d < r 2 + eaK1 1+haK1 and establish the non-existence result for suitable ranges of diffusive rates d 1 and d 2 . Proof. Let (u(x), v(x)) be any positive solution of (3) and denoteḡ = |Ω| −1 Ω gdx. By multiplying the first equation of (3) by (u −ū) and integrating over Ω, we have from Theorem 3.3 that whereL 1 is a positive constant depending on N, Ω, Λ. Similarly, multiplying the second equation of (3) by (v −v) and integrating over Ω, we have from Theorem 3.3 that where the positive constantL 2 depending on N, Ω, Λ. Thus, we obtain where L =L 1+L2

2
. Applying the well-known -Young Inequality, we obtain where is an arbitrary positive constant. It follows from the well-known Poincaré inequality that By the assumption d 2 µ 1 > r 2 − d + eaK1 1+ahK1 , we can choose > 0 small sufficiently such that d 2 µ 1 ≥ r 2 − d + eaK1 1+ahK1 + L. Taking D 1 > 1 µ1 r 1 + L , then one can conclude that u ≡ū, v ≡v if d 1 ≥ D 1 . The proof is completed.

3.3.
Existence of non-constant positive steady states. In this subsection, we show the existence of non-constant positive solutions of system (3). From Theorem 2.5, we know that the large ratio of d 2 to d 1 along with other suitable conditions will lead to the occurrence of Turing instability at a constant positive steady state of (2). The existence results of this subsection indicate that the occurrence of Turing instability at a positive constant steady state of (2) would imply the existence of non-constant positive steady state bifurcating from the constant solution.
Let X be the space defined in (7). Define

System (3) can be written as follows
where D = diag(d 1 , d 2 ) and It is easy to see that w is a positive solution of (24) if and only if w satisfies where (I − ∆) −1 is the inverse of I − ∆ subject to the zero-flux boundary condition, D −1 is the inverse of D and G(·) is a compact perturbation of the identity operator. Denote w * = E * (u * , v * ), which is the constant positive steady state of (24). We shall calculate the index of index(G(·), w * ) by the similar arguments in [26,3].
Thus, from (29)-(31), we get a contradiction. Therefore, system (3) has at least one non-constant positive solution. The proof is completed.
From the proofs of Theorem 3.6 and 1, we have the following corollary.

4.
Turing pattern formation. In Subsection 2.2, we obtain the conditions of Turing instability of the solutions to model (2). In this section, we show the Turing patterns caused by diffusion. Via numerical simulation, we find that the model dynamics exhibits spatiotemporal Turing complexity of pattern formation, including spots, strips and spots-stripes Turing patterns. We take a discrete spatial domain of size 100 × 100 (the lattice size) with the lattice constant 0.2. The numerical integration of model (2) is performed by using a finite difference approximation for the spatial derivatives and an explicit Euler method for the time integration with a time step size of 0.01. All our numerical simulations employ the zero-flux boundary conditions. The initial condition is always a small amplitude random perturbation around the positive constant steady state solution E * = (u * , v * ) of model (2). In numerical simulation, it is observed that the distributions of predator and prey are always of the same type. So, we only show our results of pattern formation to the distribution of prey u. We have taken some snapshots with red (blue) corresponding to the high (low) value of prey u.
In Fig.1, we show the time process of hot spots pattern formation of the prey u at t = 0; 1000; 5000 for the parameters as (32) and (d 1 , d 2 ) = (0.028, 0.226). It indicates that the prey population are driven by predators to a very high level in hot spots regions surrounded by areas of low prey densities. Now, taking d 2 = 0.27 and keeping other parameters unchange. In this case, µ − = 2.1499, µ + = 10.2213 and B(d 1 , d 2 ) ∩ S p = ∅. By Theorem 2.5, Turing instability emerges. We obtain the stationary stripes pattern, c.f., Fig.2.

5.
Conclusions. In this paper, we study the spatiotemporal dynamics of the reaction-diffusion predator-prey model with predator being generalist under the homogeneous Neumann boundary condition. Some basic dynamics including permanence(c.f., Theorem 2.2), non-persistnce(c.f., Theorem 2.3), the local and global stability of the nonnegative steady states of the model (2)(c.f., Theorem 2.4 and 2.6, respectively) are investigated. The conditions of Turing instability at positive constant steady states due to diffusion are given (c.f., Theorem 2.4, 2.5, and Corollary 1, respectively).
Under suitable conditions, we obtain a critical value ρ of Turing bifurcation such that if d2 d1 > ρ, then Turing bifurcation would emerge at a positive steady state of (2); while if d2 d1 < ρ, both reaction-diffusion system (2) and ODE system (1) are stability at the positive steady state. In particular, the large diffusion rate d 2 of predator or the small diffusion rate d 1 of prey will lead to the occurrence of Turing instability at the positive constant steady state of (2). From Theorem 2.3 and 3.4, we know that system (2) has no positive non-constant steady state if one of cases (1) d > r 2 + eaK1 1+haK1 , in this case species v goes extinct regardless of the diffusion   coefficients; (2) d < r 2 and either (i) ahK 1 ≤ 1 and r1 a < K2 r2 (r 2 − d), or (ii) ahK 1 > 1 and r1(1+ahK1) 2 4a 2 hK1 < K2 r2 (r 2 − d), in this case species u goes extinct regardless of the diffusion coefficients; (3) r 2 < d < r 2 + eaK1 1+haK1 , d 2 > 1 µ1 r 2 − d + eaK1 1+haK1 , and d 1 is large enough, which implies d2 d1 is small enough. Thus, in order to guarantee the existence of positive non-constant steady state, it is necessary that r 2 < d < r 2 + eaK1 1+haK1 and d2 d1 is large enough. In Theorem 3.6 and Corollary 2, this case is discussed under suitable conditions. These results indicate that the occurrence of Turing instability at a positive constant steady state of (2), along with other suitable conditions, implies the existence of non-constant positive steady state bifurcating from the constant solution.
By the numerical method, model (2) takes on some different stationary Turing patterns. For fixed d 1 = 0.028, as d 2 increases(i.e., as the ratio d2 d1 of diffusions of predator to prey increases), Turing patterns of model (2) change from spots pattern(i.e., Fig.1), stripes pattern(i.e., Fig.2) into spots-stripes pattern(i.e., Fig.3). It indicates that the pattern formation of the model (2) is rich and complex.