Global Existence and Convergence of Solutions to a Cross-Diffusion Cubic Predator-Prey System with Stage Structure for the Prey
© H. Cao and S. Fu. 2010
Received: 3 December 2009
Accepted: 30 March 2010
Published: 18 May 2010
We study a cubic predator-prey system with stage structure for the prey. This system is a generalization of the two-species Lotka-Volterra predator-prey model. Firstly, we consider the asymptotical stability of equilibrium points to the system of ordinary differential equations type. Then, the global existence of solutions and the stability of equilibrium points to the system of weakly coupled reaction-diffusion type are discussed. Finally, the existence of nonnegative classical global solutions to the system of strongly coupled reaction-diffusion type is investigated when the space dimension is less than 6, and the global asymptotic stability of unique positive equilibrium point of the system is proved by constructing Lyapunov functions.
1. Introduction and Mathematical Model
is said to be the general Lotka-Volterra predator-prey model in [1–3], and to be cubic predator-prey system in , where are the population densities of prey and predator species at time , respectively. are positive constants, is nonnegative as the intrinsic growth rate of prey population, and the sign of is undetermined. is the net mortality rate of predator population, and the survival of predator species is dependent on the survival state of prey species, and are the respective density restriction terms of prey and predator species. is the predation rate of the predator, and is the conversion rate of the predator. In , three questions about system (1.1) are discussed: the stability of nonnegative equilibrium points, and the existence, as well as numbers of limit cycle.
where and are the population densities of the immature and mature prey species, respectively, and denotes the density of the predator species. The predators live only on the immature prey species, as well as the survival of the predator species is dependent on the survival state of the immature prey species. are positive constants, and the sign of is undetermined. and are the birth rate and the mortality rate of the immature prey species, respectively. and are the net mortality rate of the mature prey population and the predator population, and is the conversion rate of the immature prey to the mature prey species. and are the respective density restriction terms of the immature prey species and predator species. is the predation rate of the predator to the immature prey population, and is the conversion rate of the predator.
where and are positive constants, and is undetermined to the sign.
where , is the unit outward normal vector of the boundary which we will assume to be smooth. The homogeneous Neumann boundary condition indicates that the above system is self-contained with zero population flux across the boundary. The positive constants , , and are said to be the diffusion coefficients, and the initial values ( ) are nonnegative smooth functions.
In the above, and are positive constants. and are the diffusion rates of the three species, respectively. are referred to as self-diffusion pressures. and are cross-diffusion pressures. The term self-diffusion implies the movement of individuals from a higher to a lower concentration region. Cross-diffusion expresses the population fluxes of one species due to the presence of the other species. Generally, the value of the cross-diffusion coefficient may be positive, negative, or zero. The term positive cross-diffusion coefficient denotes the movement of the species in the direction of lower concentration of another species, and negative cross-diffusion coefficient denotes that one species tends to diffuse in the direction of higher concentration of another species .
The main purpose of this paper is to study the asymptotic behavior of the solutions of the reaction-diffusion system (1.5) and the global existence of the solution of the cross-diffusion system (1.6). But it is necessary to denonstrate that the conclusion for the existence of global solution of system (1.6) in this paper is the generalization of the work to Lotka-Volterra competition model with cross-diffusion  and that the convergence of solution investigated in this paper which is not discussed in .
The paper will be organized as follows. In Section 2, we analyze the asymptotical stability of equilibrium points for the ODE system (1.4) via linearization and the Lyapunov method. In Section 3, we prove the global existence of solutions and the stability of the equilibrium points to the diffusion system (1.5). In Section 4, we investigate the existence of nonnegative classical global solutions by assuming , , , , , to be positive constants only for the simplicity of calculation, and the global asymptotic stability of unique positive equilibrium point to the cross-diffusion system (1.6).
2. Equilibrium Solution of the ODE System
In this section we discuss the stability of unique positive equilibrium point for system (1.4). The following theorem shows that the solution of system (1.4) is bounded.
The above is a positive constant depending only on , and further .
which implies that there exist and referring to (2.1) such that , , and .
From the comparison inequality for the ODE, we have , .
Thus the solutions for system (1.4) are bounded. Further, from the extension theorem of solutions, we have .
; (ii) , where the left equal sign holds if and only if ; (iii) ; (iv) ; (v) and , where the second equal sign holds if and only if ; (vi) and .
System (1.4) has the unique positive equilibrium point when one of the above conditions (i), (ii), (iii), (iv), (v), and (vi) holds. If holds, then is locally asymptotically stable.
Theorem 2.2 is easy to be obtained by using linearization; therefore, we omit its proof. The objective of this section is to prove the following result.
System (1.4) has the unique positive equilibrium point when one of the above conditions (i), (ii), (iii), (iv), (v), and (vi) holds. If holds, then is globally asymptotically stable.
Set . According to the Lyapunov-LaSalle invariance principle , is global asymptotic stability if inequality (2.10) and all conditions of Theorem 2.2 are satisfied. Theorem 2.3 is, thus, proved.
3. Stability of the PDE System without Cross-Diffusion
In this section, we first prove the global existence and uniform boundedness of solutions, then discuss the stability of unique positive equilibrium solution for the weakly coupled reaction-diffusion system (1.5).
Denote that , where and . It is easy to see that with . The standard PDE theory  shows that (1.5) has the unique solution , where is the maximal existence time. The following theorem shows that the solution of (1.5) is uniformly bounded, and thus .
Let be the solution of system (1.5) with initial values , and let be the maximal existence time. Then , , and , where is a positive constant depending only on and all coefficients of (1.5) and , . Furthermore, and on for any if .
Let be the solution of (1.5) with initial values . From the maximum principle for parabolic equations , it is not hard to verify that for , where is the maximal existence time of the solution . Furthermore, we know by the strong maximum principle that on for all if . Next we prove that the solution is bounded on .
for some positive constant depending only on the coefficients of (1.5). Therefore, is bounded in . Using [17, Exercise 5 of Section ], we obtain that is also bounded in . Now note that The maximum principle gives . The proof of Theorem 3.1 is completed.
Let and be positive constants. Assume that , , and is bounded from below. If , and in for some constant , then
Next we present the clear proof of the the global stability by two steps:
Step 1 (Local Stability).
For each , is invariant under the operator , and is an eigenvalue of on if and only if it is an eigenvalue of the matrix .
then , and (3.10) holds for .
Consequently, the spectrum of , consisting only of eigenvalues, lies in if (3.9) holds, and the local stability of follows [19, Theorem ].
Step 2 (Global Stability).
The global asymptotic stability of follows from (3.31) and the local stability of .
System (1.5) has the unique positive equilibrium point when one of the conditions (i), (ii), (iii), (iv), (v), and (vi) in Section 2 holds. If (3.9) and (3.21) hold, then is globally asymptotically stable.
4. Global Existence of Classical Solutions and Convergence
In this section, we discuss the existence of nonnegative classical global solutions and the global asymptotic stability of unique positive equilibrium point of system (1.6).
Some notations throughout this section are as follows: , means that for any with , , means that and are in , and with .
To obtain normal estimates of the solution for (1.6), we present a series of lemmas in the following.
yields and . It follows that there exists for any , where is a big enough positive constant such that (4.1) holds.
From (4.7), we have . Hence, . Moreover, the Sobolev embedding theorem shows that (4.6) holds.
Lemma 4.3 (Lemma can be presented by combining Lemmas and in ).
Finally, one proposes some standard embedding results which are important to obtain the normal estimates of the solution for (1.6).
Let be a fixed bounded domain and . Then for all with , one has
where is a positive constant dependent on and .
The main result about the global existence of nonnegative classical solution for the cross-diffusion system (1.6) is given as follows.
Assume that and satisfy homogeneous Neumann boundary conditions and belong to for some . Then system (1.6) has a unique nonnegative solution when the space dimension is .
where , , are Kronecker symbols.
In particular, if , then ; in other words, Theorem 4.5 is proved. For the case , from Sobolev embedding theorem, we have . Repeating the above bootstrap and Shauder estimate arguments, this completes the proof of Theorem 4.5. About space dimension , see .
where and .
In addition, is decreasing for , so we can conclude that the solution is globally asymptotically stable. The proof of Theorem 4.6 is completed.
The work of this author was partially supported by the Natural Science Foundation of Anhui Province Education Department (KJ2009B101) and the NSF of Chizhou College (XK0833) (firstname.lastname@example.org). The work of this author was partially supported by the China National Natural Science Foundation (10871160), the NSF of Gansu Province (096RJZA118), and NWNU-KJCXGC-03-47, 61 Foundations (email@example.com).
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