Qualitative analysis on a cubic predator-prey system with diffusion

In this paper, we study a cubic predator-prey model with diffusion. We first establish the global stability of the trivial and nontrivial constant steady states for the reaction diffusion system, and then prove the existence and non-existence results concerning non-constant positive stationary solutions by using topological argument and the energy method, respectively.


Introduction
Huang etc. in [6] proposed a cubic differential system, which can be considered a generalization of the predator-prey models and the mathematical form of the system satisfies the following: where X and Y represent the densities of prey and predator species at time t respectively.b 3 , b 4 , c, α, β are positive constants, and b 1 is non-negative, and the sign of b 2 is undetermined.When b 2 < 0 and b 3 = 0, the system (1.1) becomes the standard predator-prey model.The more detailed biological implication for the model, one may further refer to [6] and the references therein.
In [6], the authors introduced the following scaling transformations, and rewrite t as τ , then system (1.1) turns into where a 1 = b 1 /c, a 2 = b 2 /α, a 3 = b 3 c/α 2 and k = b 4 /β.a 1 is non-negative, and the sign of a 2 is undetermined, a 3 and k are positive constants.For system (1.2), in [6], the authors studied the properties of the equilibrium points, the existence of a uniqueness limit cycle, and the conditions for three limit cycles.
First of all, we note that (1.3) has two trivial non-negative constant steady states, namely, E 0 = (0, 0), E 1 = (u * * , 0), where u * * = (a 2 + a 2 2 + 4a 1 a 3 )/(2a 3 ).Simple analysis shows that model (1.3) has the only positive constant steady-state solution if and only if a 1 + a 2 > a 3 .We denote this steady state by (u * , v * ), where Another aspect of our goal is to investigate the corresponding steady-state problem of the reaction-diffusion system (1.3), which may display the dynamical behavior of solutions to (1.3) as time goes to infinity.This steady-state problem satisfies in Ω, It is clear that only non-negative solutions of (1.4) are of realistic interest.The remaining content in our paper is organized as follows.In section 2, we mainly analyze the global stability of constant steady states to (1.3).Then, in section 3, we give a priori estimates of upper and lower bounds for positive solutions of (1.4), and finally in section 4 we derive some non-existence and existence results of positive non-constant solutions of (1.4).
2 Some properties of solutions to (1.3) and stability of (u * , v * ) In this section, we are mainly concerned with some properties of solutions to (1.3) and the global stability of (u * , v * ) for system (1.3).Throughout this section, let (u(x, t), v(x, t)) be the unique solution of (1.3).It is easily seen that (u(x, t), v(x, t)) exists globally and is positive, namely, u(x, t), v(x, t) > 0 for all x ∈ Ω and t > 0. EJQTDE, 2011 No. 26, p. 2 2.1 Some properties of the solutions to (1.3)The following assertions characterize the global stability of each of the trivial non-negative constant steady states, and the boundedness of the positive solutions to (1.3).
Before proving the above conclusions, we need to introduce the following lemma, which can be proved using the comparison principle (see also [17]).
and the constant α > 0. Then and the constant α ≤ 0. Then lim sup In the following, we give the proof of Theorem In view of u is positive, we obtain lim t→∞ u(•, t) = 0 uniformly on Ω.For any given ε > 0 small enough, there is a From the second equation of (1.3) we have, for x ∈ Ω and t > T 1 , Thanks to Lemma 2.1 and the arbitrariness of ε > 0, it follows that lim sup Since v is also positive, we arrive at lim t→∞ v(•, t) = 0 uniformly on Ω.Before proving (ii), we firstly prove (iii).From the first equation of (1.3) we see that By Lemma 2.1, one gets lim sup For any given ε > 0, there exists By the second equation of (1.3) we have, for x ∈ Ω and t > T 2 , which asserts our result (iii).Now, we begin to verify (ii).In order to obtain the result, we need to consider two different cases.
2.2 Local stability of (u * , v * ) to system (1.3)By Theorem 2.1, from now on, without special statement, we always assume that a 1 + a 2 > a 3 , which guarantees the existence of (u * , v * ).In this subsection, we will analyze the local stability of (u * , v * ) to (1.3).To this end, we first introduce some notations.
Proof.The linearization of (1.3) at (u * , v * ) is where For each j, j = 0, 1, 2, • • • , X j is invariant under the operator L, and ξ is an eigenvalue of L on X j if and only if ξ is an eigenvalue of the matrix where detA j and TrA j are respectively the determinant and trace of A j .It is easy to check that detA j > 0 and TrA j < 0 if u * > a 2 /(2a 3 ), i.e. 4a 1 a 3 + 2k(2a 3 − a 2 ) + a 2 2 > 0. The same analysis as in [16] gives that the spectrum of L lies in {Reξ < −δ} for some positive δ independent of i ≥ 0. It is known that (u * , v * ) is uniformly asymptotically stable and the proof is complete.EJQTDE, 2011 No. 26, p. 6 2.3 Global stability of (u * , v * ) to system (1.3)In this subsection, we will be devoted to the global stability of (u * , v * ) for system (1.3).
Proof.In order to give the proof, we need to construct a Lyapunov function.First, we define We note that E(u)(t) and E(v)(t) are non-negative, E(u)(t) = 0 and E(v)(t) = 0 if and only if (u(x, t), v(x, t)) = (u * , v * ).Furthermore, easy computations yield that Similarly, Hence When u * > a 2 /a 3 , i.e., a 1 a 3 + k(a 3 − a 2 ) > 0 then dE(t)/dt ≤ 0, and the equality holds if and only if (u, v) = (u * , v * ).Hence, the standard arguments together with (iii) of Theorem 2.1 and Theorem 2.2 deduce that (u * , v * ) attracts all solutions of (1.3).This finishes the proof.EJQTDE, 2011 No. 26, p. 7 3 A priori estimates for positive solutions to (1.4) From now on, our aim is to investigate the steady-state problem (1.4).In this section, we will deduce a priori estimates of positive upper and lower bounds for positive solutions of (1.4).To this end, we first cite two known results.Lemma 3.1 (Maximum principle [8]) Suppose that g ∈ C(Ω × R).
(i) Assume that w ∈ C 2 (Ω) ∩ C 1 (Ω) and satisfies If w(x 0 ) = min Ω w, then g(x 0 , w(x 0 )) ≤ 0. Then there exists a positive constant Proof.Assume that (u, v) is a positive solution of (1.4).We set Applying Lemma 3.1 to (1.4), we obtain that From (3.1), it follows that If a 1 + a 2 > a 3 , then in view of (3.2), it is easy to see that The proof is complete.
, by Theorem 3.1 and Lemma 3.2, there exists a positive constant C 1 , such that min Now, it suffices to verify the lower bounds of v(x).We shall prove by contradiction.Suppose that Theorem 3.2 is not true, then there exists a sequence {d 2,i } ∞ i=1 with d 2,i ≥ d and the positive solution (u i , v i ) of (1.4) corresponding to By the Harnack inequality, we know that there is a positive constant C 2 independent of i such that max Moreover, integrating over Ω by parts, we have The embedding theory and the standard regularity theory of elliptic equations guarantee that there is a subsequence of (u i , w i ) also denoted by itself, and two non-negative functions u, w By (3.3) and Theorem 3.1, we have 0 < u ≤ (a 2 + a 2 2 + 4a 1 a 3 )/(2a 3 ), and when u lies in this interval, a 1 + a 2 u − a 3 u 2 ≥ 0. As a result, by the first integral identity of (3.6) we obtain u = (a 2 + a 2 2 + 4a 1 a 3 )/(2a 3 ).In view of a 1 + a 2 > a 3 , so u = (a 2 + a 2 2 + 4a 1 a 3 )/(2a 3 ) > 1 , and the second integral identity of (3.6) yields Ω wdx = 0, which implies a contradiction.This completes the proof.EJQTDE, 2011 No. 26, p. 9 4 Non-existence and existence for non-constant solutions to (1.4)

Non-existence of positive non-constant solutions
In this subsection, based on the priori estimates in Section 3 for positive solutions to (1.4), we present some results for non-existence of positive non-constant solutions of (1.3) as the diffusion coefficient d 1 or d 2 is sufficiently large.
Note that µ 1 be the smallest positive eigenvalue of the operator −∆ in Ω subject to the homogeneous Neumann boundary condition.Now, using the energy estimates, we can claim Theorem 4.1 (i) There exists a positive constant d1 = d1 (a 1 , a 2 , a 3 , k, Ω) such that (1.4) Proof.Let (u, v) be any positive solution of (1.4) and denote ḡ = (1/|Ω|) Ω g dx.Then, multiplying the corresponding equation in (1.4) by u − ū and v − v respectively, integrating over Ω, we obtain Similarly, Consequently, there exists 0 < ε ≪ 1 which depends only on a 1 , a 2 , a 3 , k, Ω , such that we yield from (4.1) that It is clear that there exists d1 depending only on a 1 , a 2 , a 3 , k, Ω, such that when µ 1 d 1 > d1 and , which asserts our result (i).
As above, we have The remaining arguments are rather similar as above.The proof is complete.

Existence of positive non-constant solutions
This subsection is concerned with the existence of non-constant positive solutions to (1.4).The main tool to be used is the topological degree theory.To set up a suitable framework where the topological degree theory can apply, let us first introduce some necessary notations.Let X be as in section 2. For simplicity, we write u = (u, v), u * = (u * , v * ).
We also denote the following sets where θ = u * (a 2 − 2a 3 u * ).Then D u G(u * ) = A.Moreover, (1.4) can be written as In order to apply the degree theory to obtain the existence of positive non-constant solutions, our first aim is to compute the index of f (d 1 , d 2 ; u) at u * .By the Leray-Schauder Theorem (see [11]), we have that if 0 is not the eigenvalue of (4.6), then where r is the number of negative eigenvalues of (4.6).
It is easy to see that, for each integer j ≥ 0, X j is invariant under D u f (d 1 , d 2 ; u * ), and ξ is an eigenvalue of D u f (d 1 , d 2 ; u * ) on X j if and only if ξ(1 + µ j ) is an eigenvalue of the matrix Thus, D u f (d 1 , d 2 ; u * ) is invertible if and only if, for all j ≥ 0, the matrix µ j I−D −1 A is nonsingular.Denote In addition, we also have that, if H(µ j ; d 1 , d 2 ) = 0, the number of negative eigenvalues of D u f (d 1 , d 2 ; u * ) on X j is odd if and only if H(µ j ; d 1 , d 2 ) < 0. Let m(µ j ) be the algebraical multiplicity of µ j .In conclusion, we can assert the following: Proposition 4.1 Suppose that, for all j ≥ 0, the matrix In fact, we observe that µ * (d We are now in the position of proving (1.4) has at least one non-constant positive solution for any d 2 ≥ d under the hypotheses of the theorem.On the contrary, suppose that this assertion is not true for some d 2 ≥ d.In the following, we will derive a contradiction by using a homotopy argument.
For such d 2 and t ∈ [0, 1], we define and consider the problem It is clear that finding positive solutions of (1.4) becomes equivalent to finding positive solutions of (4.8) for t = 1.On the other hand, for 0 ≤ t ≤ 1. u is a non-constant positive solution of (4.This contradicts (4.13).The proof is complete.Similarly, we have the following result, whose proof is similar to the above and thus is omitted.) ∈ (µ q , µ q+1 ) f or some 0 ≤ l < q, and q k=l+1 m(µ k ) is odd, then (1.4) has at least one non-constant positive solution.