Global Existence and Convergence of Solutions to a Cross-Diffusion Cubic Predator-Prey System with Stage Structure for the Prey

Abstract

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

The predator-prey model as, which follows, the ordinary differential equation system

(1.1)

is said to be the general Lotka-Volterra predator-prey model in [1–3], and to be cubic predator-prey system in [4], 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 [4], three questions about system (1.1) are discussed: the stability of nonnegative equilibrium points, and the existence, as well as numbers of limit cycle.

Referring to [5], we establish cubic predator-prey system with stage structure for the prey as follows:

(1.2)

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.

Using the scaling

(1.3)

and redenoting by , system (1.2) reduces to

(1.4)

where and are positive constants, and is undetermined to the sign.

To take into account the inhomogeneous distribution of the predators and prey in different spatial locations within a fixed bounded domain at any given time, and the natural tendency of each species to diffuse to areas of smaller population concentration, we derive the following PDE system of reaction-diffusion type:

(1.5)

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.

Note that, in recent years, there has been considerable interest to investigate the global behavior of a system of interacting populations by taking into account the effect of self as well as cross-diffusion. According to the ideas in [6–13], especially to [8, 9], the cross-diffusion term will be only included in the third equation, that is, the following cross-diffusion system:

(1.6)

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 [9].

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 [11] and that the convergence of solution investigated in this paper which is not discussed in [11].

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.

Theorem 2.1.

Let be the solution of system (1.4) with initial values , and let be the maximal existence interval of the solution. Then , where

(2.1)

The above is a positive constant depending only on , and further .

Proof.

It is easy to see that (1.4) has a unique positive local solution . Let be the maximal existence time of the solution, and combin and linearly, that is, , it follows from (1.4) that

(2.2)

Using Young inequality, we can check that there exists a positive constant depending only on and such that

(2.3)

It follows that

(2.4)

which implies that there exist and referring to (2.1) such that , , and .

Finally, we note that . Let , then

(2.5)

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 .

By the simple calculation, the sufficient conditions for system (1.4) having a unique positive equilibrium point as follows:

• (i) ; (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 .

If one of the above conditions holds, then system (1.4) has the unique positive equilibrium point , where

(2.6)

Theorem 2.2.

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.

Theorem 2.3.

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.

Proof.

We make use of the general Lyapunov function

(2.7)

where are positive constants. It holds that for any . Calculating the derivative along each solution of system (1.4), we have

(2.8)

Let and . Then

(2.9)

We observe that

(2.10)

is a sufficient condition of . So, when condition (2.10) holds, we have

(2.11)

Set . According to the Lyapunov-LaSalle invariance principle [14], 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 [15] 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 .

Theorem 3.1.

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 .

Proof.

Let be the solution of (1.5) with initial values . From the maximum principle for parabolic equations [16], 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 .

Integrating the first two equations of (1.5) over and adding the results linearly, we have that, by Young inequality,

(3.1)

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.

In order to prove the global stability of unique positive equilibrium solution for system (1.5), we first recall the following lemma which can be found in [7, 17].

Lemma 3.2.

Let and be positive constants. Assume that , , and is bounded from below. If , and in for some constant , then

Let be the eigenvalues of the operator on with the homogeneous Neumann boundary condition, and let be the eigenspace corresponding to in . Denote that , is an orthonormal basis of and . Then

(3.2)

Next we present the clear proof of the the global stability by two steps:

Step 1 (Local Stability).

Let and , where

(3.3)

The linearization of (1.5) at is

(3.4)

For each , is invariant under the operator , and is an eigenvalue of on if and only if it is an eigenvalue of the matrix .

The characteristic polynomial of is given by

(3.5)

where

(3.6)

Thus

(3.7)

where and are given by

(3.8)

According to the Routh-Hurwitz criterion [18], for each , the three roots of all have negative real parts if and only if , and . Noting that and , the three roots have negative real parts if . A direct calculation shows that is negative if

(3.9)

Now we can conclude that there exists a positive constant such that

(3.10)

In fact, let , then

(3.11)

Since as , it follows that

(3.12)

It is easy to see that are the three roots of . Thus, there exists a positive constant such that

(3.13)

By continuity, we see that there exists such that the three roots of satisfy

(3.14)

So

(3.15)

Let

(3.16)

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).

In the following, denotes a generic positive constant which does not depend on and . Let be the unique positive solution. Then it follows from Theorem 3.1 that is bounded uniformly on , that is, for all . By [20, Theorem ],

(3.17)

Define the Lyapunov function

(3.18)

Then for all . Using (1.5) and integrating by parts, we have

(3.19)

Taking and , we have that

(3.20)

where holds for

(3.21)

From Theorem 3.1 the solution of (1.5) is bounded, and so are the derivatives of and by equations in (1.5). Applying Lemma 3.2, we obtain

(3.22)

As , it follows that

(3.23)

Using inequality (3.17) and system (1.5), the derivative of is bounded in . From Lemma 3.2, we conclude that as . Therefore

(3.24)

Using the Poincaré inequality yields

(3.25)

where Thus, it follows from (3.22) and (3.25) that

(3.26)

as . So we have as . Similarly, as . Therefore, there exists a sequence with such that . As is bounded, there exists a subsequence of , still denoted by the same notation, and nonnegative constant such that

(3.27)

At , from the first equation of (1.5), we have

(3.28)

In view of (3.22) and (3.27), it follows from (3.28) that , thus

(3.29)

According to (3.17), there exists a subsequence of , denoted still by , and nonnegative functions , such that

(3.30)

In view of (3.29) and noting that in fact and , we know that . Therefore,

(3.31)

The global asymptotic stability of follows from (3.31) and the local stability of .

Theorem 3.3.

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.

Lemma 4.1.

Let be the solution of (1.6). Then there exists a positive constant (1) such that

(4.1)

Proof.

By applying the comparison principle [20] to system (1.6), we have and in . To prove that in the following, we consider the auxiliary problem

(4.2)

Notice that the functions and are sufficiently smooth in , and are quasimonotone in . Let and be a pair of upper-lower solutions for (4.2), where and are positive constants. Direct calculation with inequalities

(4.3)

yields and . It follows that there exists for any , where is a big enough positive constant such that (4.1) holds.

Lemma 4.2.

Let , and for the solution to following equation:

(4.4)

where , are positive constants and . Then there exists a positive constant , depending on and , such that

(4.5)

Furthermore,

(4.6)

Proof.

It is easy to check, from , that

(4.7)

where and . and are bounded in from (4.1). Multiplying (4.7) by , and integrating by parts over , yields

(4.8)

Using Hölder inequality and Young inequality to estimate the right side of (4.8), we have

(4.9)

with some . Substituting (4.9) into (4.8) yields

(4.10)

where depends on and . Since , the elliptic regularity estimate [10, Lemma ] yields

(4.11)

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 [11]).

Let , and let satisfy

(4.12)

and there exist positive constants and such that . Then there exists a positive constant independent of but possibly depending on , , , and such that

(4.13)

Finally, one proposes some standard embedding results which are important to obtain the normal estimates of the solution for (1.6).

Lemma 4.4.

Let be a fixed bounded domain and . Then for all with , one has

(1)

(2)

(3)

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.

Theorem 4.5.

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 .

Proof.

Step 1.

--Estimates and-Estimates of. Firstly, integrating the third equation of (1.6) over , we have

(4.14)

Thus

(4.15)

Furthermore

(4.16)

Integrating (4.14) in and moving terms yield

(4.17)

Secondly, multiplying the third equation of (1.6) by and integrating over , we have

(4.18)

Integrating the above expression in yields

(4.19)

Since from Lemma 4.2, and using Hölder inequality and Young inequality, we have

(4.20)

From (4.1) and , it holds that

(4.21)

Taking and selecting a proper such that , then applying (4.20) and (4.21) to (4.19) yields

(4.22)

Denote that . Then it follows from (4.22) that

(4.23)

It is easy to see that and for any ; hence

(4.24)

Take . Then it follows from -estimates of namely (4.15), that

(4.25)

It follows from Lemma 4.3 and (4.24) that

(4.26)

Since , is bounded by contrary proof. It follows that is bounded, that is, . It is easy to check that for all still denote by , then

(4.27)

Finally, we observe that satisfies with . So take for (4.17) and (4.19). Then there exists a positive constant such that

(4.28)

Step 2.

-Estimates of. We rewrite the third equation of (1.6) as a linear parabolic equation

(4.29)

where , , are Kronecker symbols.

To apply the maximum principle [15, Theorem , page 181] to (4.15) to obtain , we need to verify that the following conditions hold: (1) is bounded; (2) (3) , where and are positive constants, and and satisfy

(4.30)

Next we verify conditions (1)–(3) in turn. From (4.28), condition (1) is true for . One can choose such that condition (2) holds. To verify condition (3), the first equation of (1.6) is written in the divergence form

(4.31)

where is bounded in by Lemma 4.1, and for from (4.27). Application of the Hölder continuity result [15, Theorem , page 204] to (4.19) yields

(4.32)

Returning to (4.7), since for any by (4.1) and (4.27), and by (4.32), then by applying the parabolic regularity theorem [15, Theorem , pages 341-342] to (4.7) we have

(4.33)

Hence from Lemma 4.4, which shows that . Similarly, by the second equation of (1.6). Now we can show that , which imply that . In addition, obviously belongs to . It follows that one can select . Now the above three conditions are satisfied, and from [15, Theorem , page 181]. Recalling Lemma 4.1, thus there exists a positive constant for any such that

(4.34)

Step 3.

The Proof of the Classical Solutionof (1.6) infor Any. Because , we have from (4.34) that for any . So for all . It follows from [15, Lemma , page 80] that . And direct calculation yields . So we have

(4.35)

The third equation of (1.6) can be written as

(4.36)

Summarizing the above conclusions that are proved, we know that and are all bounded in . It follows from [15, Theorem page 204] that there exists such that

(4.37)

The proof of Lemma 4.2 is similar. Then we have , that is, . Applying the [13, Theorem page 204] to the second equation (1.6), there exists such that

(4.38)

Furthermore, applying Schauder estimate [15, page 320-321] yields for . Selecting and using Sobolev embedding theorem, we have . Still applying Schauder estimate, we have

(4.39)

Let . Then satisfies

(4.40)

where . By (4.35)–(4.38), we have . So applying Schauder estimate to (4.40) yields . Since , we have

(4.41)

The first equation of (1.6) can be written as

(4.42)

where . By (4.35), (4.39), and (4.41), we have . So applying Schauder estimate to (4.42) yields

(4.43)

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 [21].

Theorem 4.6.

System (1.6) has the unique positive equilibrium point when one of the conditions (i), (ii), (iii), (iv), (v), and (vi) in Section 2 holds. Let the space dimension be , and let the initial values be nonnegative smooth functions and satisfy the homogenous Neumann boundary conditions. If the following condition (4.44) holds, then the solution of (1.6) converges to in :

(4.44)

where and .

Proof.

Define the Lyapunov function

(4.45)

where and have been given in Theorem 4.6. Obviously, is nonnegative, and if and only if and . When is a positive solution of system (1.6), is well posed for all from Theorem 4.5. According to system (1.6), the time derivative of satisfies

(4.46)

It is easy to check that the final three integrands on the right side of the above expression are positive definite because of the electing of , and the sufficient and necessary conditions of the first integrand being positive definite are the following:

(4.47)

Noticing that (4.44) is the sufficient conditions of (4.47), so there exists a positive constant such that

(4.48)

Similar to the tedious calculations of , using integration by parts, Hölder inequality, and (4.34), one can verify that is bounded from above. Thus we have from (4.48) and Lemma 3.2 in Section 3 that

(4.49)

In addition, is decreasing for , so we can conclude that the solution is globally asymptotically stable. The proof of Theorem 4.6 is completed.