Bifurcation analysis of a diffusive predator – prey model in spatially heterogeneous environment

We investigate positive steady states of a diffusive predator–prey model in spatially heterogeneous environment. In comparison with the spatially homogeneous environment, the dynamics of the predator–prey model of spatial heterogeneity is more complicated. Our studies show that if dispersal rate of the prey is treated as a bifurcation parameter, for some certain ranges of death rate and dispersal rate of the predator, there exist multiply positive steady state solutions bifurcating from semi-trivial steady state of the model in spatially heterogeneous environment, whereas there exists only one positive steady state solution which bifurcates from semi-trivial steady state of the model in homogeneous environment.


Introduction
Understanding the effects of dispersal and environmental heterogeneity on the dynamics of populations is a very important and challenging topic in mathematical ecology [5].Dispersal is an important aspect of the life histories of many organisms.It allows individuals to search for resources and interact with members of their own and other species, and distribute themselves more reasonably in space, etc.The spatial heterogeneity can greatly influence the persistence, extinction and coexistence of populations, and it often give rise to certain interesting phenomena.It is demonstrated in [7] that for a Lotka-Volterra competitive model in spatially heterogeneous environment with the same resource, the slower diffuser always prevails.However, for a classical Lotka-Volterra competition system [13] with the total resource being fixed exactly at the same level, the environmental heterogeneity is usually superior to its homogeneous counterpart in the present of diffusion.Previous works [19] illustrate that for a predator-prey model in patchy environment, the spatial heterogeneity has a stabilization effects on the predator-prey interaction.There are many research results concerning the effects of dispersal and spatial heterogeneity of the environment on the dynamics of populations via predator-prey models [8,11] and competition models [4,13,14,16].
In this paper, we study a reaction-diffusion system modelling predator-prey interactions in spatially heterogeneous environment with the following form: where u(x, t) and v(x, t) denote respectively the population density of the prey and predator with corresponding migration rates µ and ν, and are required to be nonnegative.The function m(x) accounts for spatially heterogeneous carrying capacity or intrinsic growth rate of the prey population, γ is death rate of the predator.∆ := ∑ N i=1 ∂ 2 /∂x 2 i is the Laplace operator in R N (N ≥ 1) which characterizes the random motion of the predator and prey, the habitat Ω is assumed to a bounded domain in R N with smooth boundary, denoted by ∂Ω.∂u/∂n = ∇u • n, where n represents the outward unit normal vector on ∂Ω, and the homogeneous Neumann boundary condition means that no flux cross the boundary of the habitat.The reaction term is a Holling type II function response which describes the change in the density of prey attached per unit time per predator as the prey density changes.We shall assume that µ, ν, l and γ are all positive constants, u 0 and v 0 are nonnegative functions which are not identically to zero.
As was shown in [17], the joint action of migration and spatial heterogeneity can greatly influence the local dynamics of (1.1).To be more specific, in comparison with the homogeneous environment, for some certain ranges of death rate of the predator, the stability of semi-trivial steady state of (1.1) in spatially heterogeneous environment can change multiply times as the migration of the prey varies from small to large.In this paper, we would like to further investigate whether positive steady states of (1.1) can bifurcate from the semi-trivial steady state.Hence, the function m(x) is assumed to be nonconstant for reflecting the spatial heterogeneity.Throughout this paper, we shall assume that m(x) satisfies m(x) > 0, and is nonconstant and Hölder continous in Ω. (1.2) It is known [16] that under the assumption (1.2), the following logistic equation admits a unique positive solution for every µ > 0, denoted by θ(x, µ), and θ(x, µ) ∈ C 2 (Ω).
We sometimes write θ(x, µ) as θ for simplicity.By Lemma 2.3 in Section 2, the stability of semitrivial steady state (θ, 0) of (1.1) is determined by the sign of the least eigenvalue (denoted by λ 1 ) of It is well known that λ 1 is a smooth function of both µ and ν.By Lemma 2.4 in Section 2, we see that of K(µ) is more complex since θ is not necessarily monotone function with respect to µ.
To investigate more information about how (θ, 0) changes its stability as diffusion rate of the prey varies from small to large, Lou and Wang [17] further assumed that Under the assumptions (1.2) and (1.4), Lou and Wang [17] systematically investigated the stability of semi-trivial steady state (θ, 0).For five different ranges of death rate of the predator, they showed that (θ, 0) could change its stability multiply times as dispersal rate of the prey varies and obtained the following results: Theorem A ( [17]).Suppose that the nonconstant function m(x) satisfies (1.2), then the following conclusions hold.
Remark 1.1.From Theorem A, we see that Cases (i) and (v) can not have bifurcation from semi-trivial steady state (θ, 0).Therefore, it suffices to investigate Cases (ii), (iii) and (iv) in this paper.For these three cases, we have the following statements.
In view of Theorem A and Remark 1.1, we are able to apply bifurcation theory to inquire how many positive solutions which can bifurcate from semi-trivial steady state (θ, 0).Furthermore, we can investigate local stability of the bifurcating solutions.Our main conclusions of this paper are the following Theorems 1.2 and 1.3.If dispersal rate of the prey µ is treated as a bifurcation parameter, we have the following conclusions: Theorem 1.2.Suppose that m(x) satisfies (1.2), then the following conclusions hold.
(a) If γ 1 < γ < γ 2 , for every ν > ν, there exists some small δ 1 > 0 such that a branch of steady state solution (u If dispersal rate of the predator ν is regarded as a bifurcation parameter, we also have the corresponding results.Theorem 1.3.Suppose that m(x) satisfies (1.2), then the following conclusions hold.
(b) If γ 2 < γ < γ 3 and m(x) satisfies (1.4) as well, for small or large µ, there exists some small ρ 2 > 0 such that two branches of steady state solutions , respectively, and they can be parameterized by , respectively, and the branch of steady state solutions to and m(x) satisfies (1.4) as well, for small µ, there exists some small ρ 3 > 0 such that a branch of steady state solution (u 4 * , v 4 * ) to (1.1) bifurcates from (θ, 0) at ν = ν * 4 , and it can be parameterized by ν for the range ν ∈ (ν * 4 − ρ 3 , ν * 4 ).Furthermore, the bifurcating solution and the branch of steady state solutions to (1.1) bifurcating from (ν * 4 , θ, 0) extends to zero in ν.For predator-prey models in spatially homogeneous environment, there have been many works concerning the local or global bifurcation results [1,2,9,10,21], we here use bifurcation theory to examine a predator prey model in spatial heterogeneity of the environment and demonstrate that positive steady state solutions could bifurcate from semi-trivial steady state of the model.Theorem 1.3 tells us that the bifurcation branch of positive solutions to (1.1) can be extended from (ν * i , θ, 0) (i = 1, 2, 3, 4) to zero in ν.However, it is quite difficult to extend the results of Theorem 1.2 to global bifurcation.One of the main reasons is that the limit behavior of positive steady states as dispersal rate of the prey approaches to zero is not clear.A deep understanding of the limit behavior of positive steady states of the model with small dispersal rate seems to be a very interesting and challenging problem, awaiting for further investigation.
The rest of this paper is organized as follows: In Section 2 we present Lemmas 2.1-2.4.Section 3 is devoted to the proof of Lemmas 3.1-3.9,Theorems 1.2, 1.3 and Theorem 3.10.

Preliminaries
In this section, we will present several lemmas which shall be used in subsequence analysis.(ii) For any µ > 0, max 3) and the positivity of θ, we see that zero is the smallest eigenvalue of the operator −µ∆ − (m − θ).By the comparison principle for eigenvalues and the positivity of θ, the smallest eigenvalue of the operator F u (µ, θ) is strictly positive, hence F u (µ, θ) is invertible.By the implicit function theorem [5], θ(x, µ) is differentiable with respect to µ.
(ii) λ 1 satisfies the following properties: Proof.The smooth dependence of λ 1 on ν can be found in [5].Part (i) can be established by the variational characterization of λ 1 .Part (ii) can be proved by using Part (i) of Lemma 2.1, we skip it here.

Local bifurcation of steady states
In this section, by applying local bifurcation theory [6,20], we will choose dispersal rates of the prey and predator as bifurcation parameters, respectively, and prove its corresponding local bifurcation conclusions.To this end, we write positive steady states of (1.1) as: We observe that F(µ, θ, 0) = 0 and the derivatives D µ F(µ, u, v), D (u,v) F(µ, u, v) and D µ D (u,v) F(µ, u, v) exist and are continuous close to (µ, θ, 0).
We separate the following proof into two cases.
Dividing the equation of ψ in (2.1), integrating by parts and after some reorganization, we have Hence, for any µ > µ * , we conclude λ 1 < 0 for any ν > 0. For every 2) and lim ν→∞ λ 1 = γ − K(µ) > 0, by Lemma 2.4, we see that there exists a unique ν * Hence, there exists some function i.e., λ 1 = 0 is the smallest eigenvalue of (2.1) with ν = ν * 1 and ψ * 1 is its corresponding eigenfunction.Since it is easy to testify that the kernel of where ψ * 1 is the unique positive solution of (3.12) up to a constant multiplier, and ϕ * 1 is uniquely determined by Furthermore, it follows from the Fredholm alternative that codim R(D (u,v) G| (ν * 1 ,θ,0) ) = 1.For the transversality condition, (3.11) into the equation of v and dividing both sides by s, we have
For convenience, we split the following proof into two cases.

Proof. Since lim
1+max Ω m < 0 as µ → 0 (by Lemma 2.1), and By similar argument to that of Lemma 3.5, we still can check that the transversality condition holds here.Moreover, we have ν 4 (0) < 0.
Proof.(a) Applying the maximum principle to the equation of u in (3.1), we have max x∈Ω u ≤ max x∈Ω m.Then the boundedness of u follows.Integrating the equations of u and v in (3.1), applying the boundary condition and after some rearrangement, we have Hence, (3.17) By Harnack inequality [15] and the equation of v in (3.1), we see that there exists some constant . This together with (3.17) implies the second inequality of (3.16).
The proof of Theorem 1.3.Local bifurcation results of Case (a) follows from Lemma 3.5, the linear stability of the bifurcating solution follows from Lemma 3.6.While local bifurcation results of Cases (b) and (c) can be found in Lemmas 3.7 and 3.8, and their linear stability of the bifurcating solution can be proved by similar argument to that of Case (a).Now it remains to investigate whether local bifurcation conclusions can be extended to global one, our following proof shows that it is true.By Lemma 3.9 and global bifurcation theory [20], we see that the bifurcating solution (ν, u * 1 , v * 1 ) can be extended from (ν * 1 , θ, 0) to zero in ν.A similar argument yields that global bifurcation conclusions of Cases (b) and (c) also hold.
In contrast with the heterogeneous environment, if m is a positive constant, then (1.1) has a semi-trivial steady state (m, 0) which is independent of µ.For this case, we can choose m as a bifurcation parameter, utilize local bifurcation theory and obtain the following theorem.Theorem 3.10.Suppose that m is constant and γ < l.For any µ, ν > 0, there exists some small δ > 0, some function m(s) ∈ C 2 (−δ, δ) with m(0) = m * such that all nonnegative steady state solutions of (1.1) close to (m * , m, 0) can be parameterized as (m, u, v) = (m(s), m + sϕ 1 + s 2 ϕ * 1 (s), where ψ 1 is some positive constant, ϕ 1 is determined by (3.18), and (ϕ * 1 (s), ψ * 1 (s)) lies in the complement of the kernel of D (u,v) F| (m * ,m,0) in X.Moreover, the bifurcation direction of the solution (m * , m, 0) can be characterized by m (0) > 0. In addition, the bifurcating solution (m, u, v) is locally stable for small s.
Proof.By Lemma 2.3, the stability of (m, 0) is determined by the sign of the smallest eigenvalue (denoted as λ 1 ) of the eigenvalue problem:

3 )
and the positivity of θ, we see that zero is the smallest eigenvalue of the operator −µ * 1 ∆ − (m − θ(x, µ * 1 )) with homogeneous Neumann boundary condition.By the comparison principle for eigenvalues and the positivity of θ, the smallest eigenvalue of the operator −µ * 1 ∆ − (m − 2θ(x, µ * 1 )) with homogeneous Neumann boundary condition is strictly positive, thus

3 )
and the positivity of θ, we see that zero is the smallest eigenvalue of the operator −µ∆ − (m − θ) with homogeneous Neumann boundary condition.By the comparison principle for eigenvalues, the smallest eigenvalue of the operator −µ∆ − (m − 2θ) with homogeneous Neumann boundary condition is strictly positive, hence