This paper deals with a Holling type III diffusive predatorprey model with stage structure and nonlinear density restriction in the space . We first consider the asymptotical stability of equilibrium points for the model of ODE type. Then, the existence and uniform boundedness of global solutions and stability of the equilibrium points for the model of weakly coupled reactiondiffusion type are discussed. Finally, the global existence and the convergence of solutions for the model of crossdiffusion type are investigated when the space dimension is less than 6.
1. Introduction
Population models with stage structure have been investigated by many researchers, and various methods and techniques have been used to study the existence and qualitative properties of solutions [1–9]. However, most of the discussions in these works are devoted to either systems of ODE or weakly coupled systems of reactiondiffusion equations. In this paper we investigate the global existence and convergence of solutions for a strongly coupled crossdiffusion predatorprey model with stage structure and nonlinear density restriction. Nonlinear problems of this kind are quite difficult to deal with since the usual idea to apply maximum principle arguments to get priori estimates cannot be used here [10].
Consider the following predatorprey model with stagestructure:
where , denote the density of the immature and mature population of the prey, respectively, is the density of the predator. For the prey, the immature population is nonlinear density restriction. is assumed to consume with Holling type III functional response and contributes to its growth with rate . For more details on the backgrounds of this model see references [11, 12].
Using the scaling and redenoting by , we can reduce the system (1.1) to
where
To take into account the natural tendency of each species to diffuse, we are led to the following PDE system of reactiondiffusion type:
where is a bounded domain in with smooth boundary , is the outward unit normal vector on , and . are nonnegative smooth functions on . The diffusion coefficients are positive constants. The homogeneous Neumann boundary condition indicates that system (1.3) is selfcontained with zero population flux across the boundary. The knowledge for system (1.3) is limited (see [13–17]).
In the recent years there has been considerable interest to investigate the global behavior for models of interacting populations with linear density restriction by taking into account the effect of selfas well as crossdiffusion [18–26]. In this paper we are led to the following crossdiffusion system:
where are the diffusion rates of the three species, respectively. are referred as selfdiffusion pressures, and is crossdiffusion pressure. The term selfdiffusion implies the movement of individuals from a higher to a lower concentration region. Crossdiffusion expresses the population fluxes of one species due to the presence of the other species. The value of the crossdiffusion coefficient may be positive, negative, or zero. The term positive crossdiffusion coefficient denotes the movement of the species in the direction of lower concentration of another species and negative crossdiffusion coefficient denotes that one species tends to diffuse in the direction of higher concentration of another species [27]. For , problem (1.4) becomes strongly coupled with a full diffusion matrix. As far as the authors are aware, very few results are known for crossdiffusion systems with stagestructure.
The main purpose of this paper is to study the asymptotic behavior of the solutions for the reactiondiffusion system (1.3), the global existence, and the convergence of solutions for the crossdiffusion system (1.4). The paper will be organized as follows. In Section 2 a linear stability analysis of equilibrium points for the ODE system (1.2) is given. In Section 3 the uniform bound of the solution and stability of the equilibrium points to the weakly coupled system (1.3) are proved. Section 4 deals with the existence and the convergence of global solutions for the strongly coupled system (1.4).
2. Global Stability for System (1.2)
Let . If , then (1.2) has semitrivial equilibria , where . To discuss the existence of the positive equilibrium point of (1.2), we give the following assumptions:
where . Let one curve : , and the other curve : . Obviously, passes the point . Noting that attains its maximum at , thus when , . has the asymptote and passes the point . In this case, and have unique intersection , as shown in Figure 1. is the unique positive equilibrium point of (1.2), where , , . In addition, the restriction of the existence of the positive equilibrium can be removed, if .
The Jacobian matrix of the equilibrium is
The characteristic equation of () is . is a saddle for . In addition, the dimensions of the local unstable and stable manifold of are 1 and 2, respectively. is locally asymptotically stable for .
The Jacobian matrix of the equilibrium is
where . The characteristic equation of () is , where
According to RouthHurwitz criterion, is locally asymptotically stable for and , that is, and .
The Jacobian matrix of the equilibrium is
where
The characteristic equation of () is , where
According to RouthHurwitz criterion, is locally asymptotically stable for . Obviously, can be checked by (2.1).
Now we discuss the global stability of equilibrium points for (1.2).
Theorem 2.1.
(i) Assume that (2.1),
hold, then the equilibrium point of (1.2) is globally asymptotically stable.
(ii) Assume that , and hold, then the equilibrium point of (1.2) is globally asymptotically stable.
(iii) Assume that holds, then the equilibrium point of (1.2) is globally asymptotically stable.
Proof.
(i) Define the Lyapunov function
Calculating the derivative of along the positive solution of (1.2), we have
When , the minimum of and is and 0, respectively; the maximum of is are and , respectively. Thus, when (2.8) hold, According to the LyapunovLaSalle invariance principle [28], is globally asymptotically stable if (2.1)–(2.3) hold.
(ii) Let
Then
Noting that the maximum of is , and , we find . Therefore,
(iii) Let
then
Thus, for . This completes the proof of Theorem 2.1.
3. Global Behavior of System (1.3)
In this section we discuss the existence, uniform boundedness of global solutions, and the stability of constant equilibrium solutions for the weakly coupled reactiondiffusion system (1.3). In particular, the unstability results in Section 2 also hold for system (1.3) because solutions of (1.2) are also solutions of (1.3).
Theorem 3.1.
Let be nonnegative smooth functions on . Then system (1.3) has a unique nonnegative solution , and
on . In particular, if , then for all .
Proof.
It is easily seen that is sufficiently smooth in and possesses a mixed quasimonotone property in . In addition, and are a pair of lowerupper solutions of problem (1.3) (cf. in (3.1)). From [29, Theorem 5.3.4], we conclude that (1.3) exists a unique classical solution satisfying (3.1). According to strong maximum principle, it follows that . So the proof of the Theorem is completed.
Remark 3.2.
When (namely ), system (1.3) reduces to a system in which the immature population of the prey is linear density restriction. Similar to the proof of Theorem 3.1, we have
Now we show the local and global stability of constant equilibrium solutions for (1.3), respectively.
Theorem 3.3.
(i) Assume that (2.1) holds, then the equilibrium point of (1.3) is locally asymptotically stable.
(ii) Assume that , , and hold, then the equilibrium point of (1.3) is locally asymptotically stable.
(iii) Assume that holds, then the equilibrium point of (1.3) is locally asymptotically stable.
Proof.
Let be the eigenvalues of the operator on with Neumann boundary condition, and let be the eigenspace corresponding to in . Let
where is an orthonormal basis of , then
Let , , where
The linearization of (1.3) is at . For each , is invariant under the operator L, and is an eigenvalue of L on , if and only if is an eigenvalue of the matrix . The characteristic equation is , where
From RouthHurwitz criterion, we can see that three eigenvalues (denoted by , , ) all have negative real parts if and only if . Noting that , we must have . It is easy to check that if (see Section 2).
We can conclude that there exists a positive constant , such that
In fact, let , then
Since as , it follows that
Clearly, has the three roots . Let . By continuity, there exists such that the three roots of satisfy
Let , then . Let , then (3.7) holds. According to [30, Theorem 5.1.1], we have the locally asymptotically stability of .
(ii) The linearization of (1.4) is at , where , and
The characteristic equation of is , where
The three roots of all have negative real parts for and . Namely, is the locally asymptotically stable, if and .
(iii) The linearization of (1.3) is at , where , and
Similar to (i), is locally asymptotically stable, when .
Remark 3.4.
When denote . If , then (1.3) has the semitrivial equilibrium point , where . If , then (1.3) has a unique positive equilibrium point . Similar as Theorem 3.3, we have the following.
(i)If , , and (namely, , , ), then is locally asymptotically stable.
(ii)If and , then is locally asymptotically stable.
(iii)If , then is locally asymptotically stable.
Before discussing the global stability, we give an important lemma which has been proved in [31, Lemma ] or in [32, Lemma ].
Lemma 3.5.
Let be positive constants. Assume that , , and is bounded from below. If and for some positive constant , then
Theorem 3.6.
(i) Assume that (2.1),
hold, then the equilibrium point of system (1.3) is globally asymptotically stable.
(ii) Assume that , and hold, then the equilibrium point of system (1.3) is globally asymptotically stable.
(iii)Assume that and hold, then the equilibrium point of system (1.3) is globally asymptotically stable.
Proof.
Let be the unique positive solution of (1.3). By Theorem 3.1, there exists a positive constant C which is independent of and such that , for . By [33, Theorem ],
(i) Define the Lyapunov function
By Theorem 3.1, is defined well for all solutions of (1.3) with the initial functions . It is easily see that and if and only if .
Calculating the derivative of along positive solution of (1.3) by integration by parts and the Cauchy inequality, we have
It is not hard to verify that
if (3.14) hold. Applying Lemma 3.5, we can obtain
Recomputing , we find
From (3.15), we can see that is bounded in , . It follows from Lemma 3.5 and (3.15) that as . Namely,
Using the Pioncaré inequality, we have
where Noting that
according to (3.19) and (3.22), we can see
Thus, there exists as . Applying the boundness of , there exists a subsequence of , denoted still by , such that On the one hand
On the other hand
According to (3.19) to compute the limit of the previous equation and using the uniqueness of the limit, we have , and
It follows from (3.15) that there exists a subsequence of , denoted still by , and nonnegative functions , such that
Applying (3.19)–(3.27), we obtain that , and
In view of Theorem 3.3, we can conclude that is globally asymptotically stable.
(ii) Let
Then
Therefore, It follows that the equilibrium point of (1.3) is globally asymptotically stable.
(iii) Define
Then
When ,
The following proof is similar to (i).
Remark 3.7.
When , Theorem 3.6 shows the following.
(i) Assume that ,
hold, then the equilibrium point of (1.3) is globally asymptotically stable.
(ii) Assume that hold, then the equilibrium point of (1.3) is globally asymptotically stable.
(iii) Assume that and hold, then the equilibrium point of (1.3) is globally asymptotically stable.
Example 3.8.
Consider the following system:
Using the software Matlab, one can obtain , . It is easy to see that the previous system satisfies the all conditions of Theorem 3.6(i). So the positive equilibrium point (0.5637,0.5637,0.1199) of the previous system is globally asymptotically stable.
4. Global Existence and Stability of Solutions for the System (1.4)
By [34–36], we have the following result.
Theorem 4.1.
If , then (1.4) has a unique nonnegative solution , where is the maximal existence time of the solution. If the solution satisfies the estimate
then . If, in addition, , then
In this section, we consider the existence and the convergence of global solutions to the system (1.4).
Theorem 4.2.
Let and the space dimension . Suppose that are nonnegative functions and satisfy zero Neumann boundary conditions. Then (1.4) has a unique nonnegative solution
In order to prove Theorem 4.2, some preparations are collected firstly.
Lemma 4.3.
Let be a solution of (1.4). Then
where .
Proof.
From the maximum principle for parabolic equations, it is not hard to verify that and is bounded.
Multiplying the second equation of (1.4) by , adding up the first equation of (1.4), and integrating the result over , we obtain
Using Young inequality and Hlder inequality, we have
where It follows from (4.3) and (4.4) that
Thus,
where depends on and coefficients of (1.4). In addition, there exists a positive constant , such that
Integrating the first equation of (1.4) over , we have
Integrating (4.8) from to , we have
According to (4.7), there exists a positive constant , such that
Multiplying the second equation of (1.4) by and integrating it over , we obtain
Integrating the previous inequation from to , we have
Lemma 4.4.
Let be a solution of (1.4), , and . Then there exists a positive constant depending on and , such that
Furthermore and
Proof.
satisfies the equation
where are functions of and so are bounded because of Lemma 4.3.
Multiply the second equation of (1.4) by and integrate it over to obtain
Then
and . From a disposal similar to the proof of Lemma 2.2 in [23], we have . Using a standard embedding result, we obtain
Lemma 4.5 (see [23, Lemmas 2.3 and 2.4]).
Let , , and let be any number which may depend on . Then there is a constant depending on , and such that
for any with for all .
To obtain estimates of , we establish estimates of in the following lemma.
Lemma 4.6.
Let , , then there exist positive constants and , such that
Proof.
Multiply the first equation of (1.4) by for and integrate by parts over to obtain
Integrating (4.19) from 0 to , we have
Then substitution of , into (4.20) leads to
It follows from Hlder inequality and Lemma 4.3 that
Note that , and for . From Hlder inequality, Young inequality, and Lemma 4.4, we have
Substitution of (4.22) and (4.23) into (4.21) leads to
where is arbitrary and .
Choose such that
then it follows from (4.24) that
Let
Then for
According to Lemma 4.5 and the definition of , we can see
Combining (4.26) and (4.29), we have
where . Therefore is bounded from (4.30).
From (4.29), we have . Namely, , . Combining (4.28), we have , where .
Setting in (4.20) (it is easily checked that , i.e., ), we have .
Multiplying the second equation of (1.4) by and integrating it over , we have
The result of can be obtained from an analogue of the previous proof of 's.
Lemma 4.7.
Let , then there exists a positive constant such that
Proof.
We will prove this lemma by [37, Theorem 7.1, page 181]. At first, we rewrite the first two equations of (1.4) as
where , , , is symbol. It follows from Lemma 4.6 that , .
By the third equation of (1.4), we have
It follows from Lemma 4.3 that is bounded in . Applying Theorem [37, Page 204] to (4.34), we have
Recall that satisfy (4.14) in Lemma 4.4, that is,
where is bounded. Since by (4.35), applying Theorem [37, page 341342] to (4.36), we have
It follows from [37, Lemma , page 80] that and so . Recall from Lemma 4.6 that , so that by applying Theorem [37, Page 181] to (4.33).
Proof of Theorem 4.2.
Firstly, Theorem 4.2 can be proved in a similar way as Theorem in [21, 25] when the space dimension .
Secondly, for , applying Lemma [37, Page 80] to (4.36), we have
Since , we obtain
The first two equations can be written in the divergence form as
where . It follows from Lemmas 4.1, 4.5, and (4.39) that are bounded. Thus applying Theorem [37, Page 204] to (4.40) leads to
We rewrite the third equation of (1.4) as
where . Applying Schauder estimate [29, Theorem , page 114] to (4.42) gives
Let
then
where , . From (4.41), we have . It follows from (4.41) and (4.43) that . Applying Schauder estimate to (4.45) gives
Solving equations (4.44) for , respectively, we have
In particular, to conclude , we need to repeat the above bootstrap technique. Since is arbitrary, so the classical solution of (1.4) exists globally in time.
Now we discuss the global stability of the positive equilibrium (see Section 2) for (1.4).
Theorem 4.8.
Assume that the all conditions in Theorem 4.2, (2.1), and
hold. Let be the unique positive equilibrium point of (1.4), and let be a positive solution for (1.4). Then
provided that is large enough.
Proof.
Define the Lyapunov function
Let be a positive solution of (1.4), Then
The first integrand in the right hand of the previous inequality is positive definite if
Therefore, when the all conditions in Theorem 4.8 hold, there exists a positive constant such that
This implies that . So the proof of Theorem 4.8 is completed.
Acknowledgments
This work has been partially supported by the China National Natural Science Foundation (no. 10871160), the NSF of Gansu Province (no. 096RJZA118), the Scientific Research Fund of Gansu Provincial Education Department, and NWNUKJCXGC0347 Foundation.
References

Aiello, WG, Freedman, HI: A timedelay model of singlespecies growth with stage structure. Mathematical Biosciences. 101(2), 139–153 (1990). PubMed Abstract  Publisher Full Text

Zhang, X, Chen, L, Neumann, AU: The stagestructured predatorprey model and optimal harvesting policy. Mathematical Biosciences. 168(2), 201–210 (2000). PubMed Abstract  Publisher Full Text

Liu, S, Chen, L, Liu, Z: Extinction and permanence in nonautonomous competitive system with stage structure. Journal of Mathematical Analysis and Applications. 274(2), 667–684 (2002). Publisher Full Text

Lin, Z: Time delayed parabolic system in a twospecies competitive model with stage structure. Journal of Mathematical Analysis and Applications. 315(1), 202–215 (2006). Publisher Full Text

Xu, R: A reactiondiffusion predatorprey model with stage structure and nonlocal delay. Applied Mathematics and Computation. 175(2), 984–1006 (2006). Publisher Full Text

Xu, R, Chaplain, MAJ, Davidson, FA: Global convergence of a reactiondiffusion predatorprey model with stage structure for the predator. Applied Mathematics and Computation. 176(1), 388–401 (2006). Publisher Full Text

Xu, R, Chaplain, MAJ, Davidson, FA: Global convergence of a reactiondiffusion predatorprey model with stage structure and nonlocal delays. Computers & Mathematics with Applications. 53(5), 770–788 (2007). PubMed Abstract  Publisher Full Text

Wang, M: Stability and Hopf bifurcation for a preypredator model with preystage structure and diffusion. Mathematical Biosciences. 212(2), 149–160 (2008). PubMed Abstract  Publisher Full Text

Wang, Z, Wu, J: Qualitative analysis for a ratiodependent predatorprey model with stage structure and diffusion. Nonlinear Analysis: Real World Applications. 9(5), 2270–2287 (2008). Publisher Full Text

Galiano, G, Garzón, ML, Jüngel, A: Semidiscretization in time and numerical convergence of solutions of a nonlinear crossdiffusion population model. Numerische Mathematik. 93(4), 655–673 (2003). Publisher Full Text

Chen, L: Mathematical Models and Methods in Ecology, Science Press, Beijing, China (1988)

Chen, LJ, Sun, JH: The uniqueness of a limit cycle for a class of Holling models with functional responses. Acta Mathematica Sinica. 45(2), 383–388 (2002)

Li, WT, Wu, SL: Traveling waves in a diffusive predatorprey model with Holling typeIII functional response. Chaos, Solitons & Fractals. 37(2), 476–486 (2008). PubMed Abstract  Publisher Full Text

Ko, W, Ryu, K: Qualitative analysis of a predatorprey model with Holling type II functional response incorporating a prey refuge. Journal of Differential Equations. 231(2), 534–550 (2006). Publisher Full Text

Zhang, H, Georgescu, P, Chen, L: An impulsive predatorprey system with BeddingtonDeAngelis functional response and time delay. International Journal of Biomathematics. 1(1), 1–17 (2008). Publisher Full Text

Fan, Y, Wang, L, Wang, M: Notes on multiple bifurcations in a delayed predatorprey model with nonmonotonic functional response. International Journal of Biomathematics. 2(2), 129–138 (2009). Publisher Full Text

Wang, F, An, Y: Existence of nontrivial solution for a nonlocal elliptic equation with nonlinear boundary condition. Boundary Value Problems. 2009, (2009)

Lou, Y, Ni, WM: Diffusion, selfdiffusion and crossdiffusion. Journal of Differential Equations. 131(1), 79–131 (1996). Publisher Full Text

Lou, Y, Ni, WM, Wu, Y: On the global existence of a crossdiffusion system. Discrete and Continuous Dynamical Systems. 4(2), 193–203 (1998)

Shim, SA: Uniform boundedness and convergence of solutions to crossdiffusion systems. Journal of Differential Equations. 185(1), 281–305 (2002). Publisher Full Text

Shim, SA: Uniform boundedness and convergence of solutions to the systems with crossdiffusions dominated by selfdiffusions. Nonlinear Analysis: Real World Applications. 4(1), 65–86 (2003). Publisher Full Text

Choi, YS, Lui, R, Yamada, Y: Existence of global solutions for the ShigesadaKawasakiTeramoto model with weak crossdiffusion. Discrete and Continuous Dynamical Systems. 9(5), 1193–1200 (2003)

Choi, YS, Lui, R, Yamada, Y: Existence of global solutions for the ShigesadaKawasakiTeramoto model with strongly coupled crossdiffusion. Discrete and Continuous Dynamical Systems. 10(3), 719–730 (2004)

Pang, PYH, Wang, MX: Existence of global solutions for a threespecies predatorprey model with crossdiffusion. Mathematische Nachrichten. 281(4), 555–560 (2008). Publisher Full Text

Fu, S, Wen, Z, Cui, S: Uniform boundedness and stability of global solutions in a strongly coupled threespecies cooperating model. Nonlinear Analysis: Real World Applications. 9(2), 272–289 (2008). Publisher Full Text

Yang, F, Fu, S: Global solutions for a tritrophic food chain model with diffusion. The Rocky Mountain Journal of Mathematics. 38(5), 1785–1812 (2008). Publisher Full Text

Dubey, B, Das, B, Hussain, J: A predatorprey interaction model with self and crossdiffusion. Ecological Modelling. 141(1–3), 67–76 (2001)

Hale, JK: Ordinary Differential Equations, Krieger, Malabar, Fla, USA (1980)

Ye, Q, Li, Z: Introduction to ReactionDiffusion Equations, Science Press, Beijing, China (1999)

Henry, D: Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, Springer, Berlin, Germany (1993)

Lin, Z, Pedersen, M: Stability in a diffusive foodchain model with MichaelisMenten functional response. Nonlinear Analysis: Theory, Methods & Applications. 57(3), 421–433 (2004). PubMed Abstract  Publisher Full Text

Wang, M: Nonliear Parabolic Equation of Parabolic Type, Science Press, Beijing, China (1993)

Brown, KJ, Dunne, PC, Gardner, RA: A semilinear parabolic system arising in the theory of superconductivity. Journal of Differential Equations. 40(2), 232–252 (1981). Publisher Full Text

Amann, H: Dynamic theory of quasilinear parabolic equations. I. Abstract evolution equations. Nonlinear Analysis: Theory, Methods & Applications. 12(9), 895–919 (1988). PubMed Abstract  Publisher Full Text

Amann, H: Dynamic theory of quasilinear parabolic equations. II. Reactiondiffusion systems. Differential and Integral Equations. 3(1), 13–75 (1990)

Amann, H: Dynamic theory of quasilinear parabolic systems. III. Global existence. Mathematische Zeitschrift. 202(2), 219–250 (1989). Publisher Full Text

Ladyženskaja, OA, Solonnikov, VA, Ural'ceva, NN: Linear and Quasilinear Equations of Parabolic Type, Translations of Mathematical Monographs,p. xi+648. American Mathematical Society, Providence, RI, USA (1967)