Global Behavior of Solutions in a Predator-Prey Cross-Diffusion Model with Cannibalism

The global asymptotic behavior of solutions in a cross-diffusive predator-prey model with cannibalism is studied in this paper. Firstly, the local stability of nonnegative equilibria for the weakly coupled reaction-diffusion model and strongly coupled cross-diffusion model is discussed. It is shown that the equilibria have the same stability properties for the corresponding ODE model and semilinear reaction-diffusion model, but under suitable conditions on reaction coefficients, cross-diffusion-driven Turing instability occurs. Secondly, the uniform boundedness and the global existence of solutions for the model with SKT-type cross-diffusion are investigated when the space dimension is one. Finally, the global stability of the positive equilibrium is established by constructing a Lyapunov function. The result indicates that, under certain conditions on reaction coefficients, the model has no nonconstant positive steady state if the diffusion matrix is positive definite and the self-diffusion coefficients are large enough.


Introduction
In 1999, Magnússon [1] proposed the following predatorprey model: where X, Y, and Z represent the biomasses of adult predators, juvenile predators, and prey, respectively, M mat and M juv represent the death rates of adult predators and juvenile predators, respectively, A denotes the specific rate of juveniles recruited to the adults class, R denotes the birth rate, T and U are logistic coefficients, S represents the cannibalism attack rate, B � cS, where c is the conversion efficiency of eaten juveniles into adult biomass, and the biological meanings of V and C are similar to S and B. For more details on model (1), please refer to [1].
and still denote t by t; then, (1) becomes We can see that e is the only parameter depending on S in system (3).
Most works on cannibalism adopt a McKendrick-von Foerster age-structured model [2][3][4], but the model is usually rewritten as an ODE system or nonlinear Volterra integral equations [4,5]. In 1995, Kohlmeier and Ebenhöh [6] studied a two-dimensional ODE model without any structure, and they obtained that, in some cases, cannibalism can lead to a higher long term predator stock size. en, van den Bosch and Gabriel [7] found that cannibalism can stabilize a predator-prey system in a structured model where the oscillations are due to age structure.
Magnússon [1] discussed model (1) which is in some ways a simplification of the system studied by van den Bosch and Gabriel in [7]. Magnússon made two simplifications. Firstly, it is assumed that all juveniles are vulnerable to predation by the adults. Secondly, instantaneous maturation into the adult class is proportional to the present juvenile biomass, i.e., a constant per capita rate of maturation, and making this simplifying assumption means that any oscillations that may occur are not caused by a delay inherent in the system. Moreover, the stability and Hopf bifurcation of solutions for system (3) were studied in [1]. e author obtained the following conclusions. On one hand, if the mortality rate of juveniles is low and/or the recruitment rate to the mature population is high, then there is a stable equilibrium with all three population sizes as positive. On the other hand, if the mortality rate of juveniles is high and/ or the recruitment rate to the mature population is low, then the equilibrium will be stable for low levels of cannibalism, but a loss of stability by a Hopf bifurcation will take place as the level of cannibalism increases.
As we all know, the growth of biological population depends not only on time, but also on spatial distribution. Spatial species interaction includes the self-diffusion, which is the natural dispersive force of movement of individual, and the cross-diffusion, that is, the population fluxes of one species due to the presence of other species [8][9][10][11][12][13][14][15][16]. erefore, taking into account the effect of competition for resources on species growth law, we are naturally led to the following weakly coupled reaction-diffusion model: and the Shigesada-Kawasaki-Teramoto model with strongly coupled cross-diffusion where Ω ⊂ R n is a bounded domain with smooth boundary zΩ and η is the outward unit normal vector of the boundary zΩ, z η � z/zη. e coefficients d i (i � 1, 2, 3) are the diffusion rates of u, v, and w, α ii (i � 1, 2, 3) are referred as selfdiffusion pressures, and α ij (i ≠ j, i, j � 1, 2, 3) are crossdiffusion coefficients which may be positive, negative, or zero [17]. e functions u 0 , v 0 , and w 0 are nonnegative functions which are not identically zero.
Very recently, Zhang et al. [18] found that the positive equilibrium of (3) with r � 0 can undergo stability switch (from stable to unstable to stable, or from unstable to stable to unstable) with the change of the cannibalization rate.

Complexity
eir results indicate that large cannibalization rate can make the positive equilibrium globally stable although its stability would change with the increase of the rate. In fact, an important research subject for cannibalism models is the stabilizing/destabilizing effect of cannibalism [1,6,19]. For this, we need to compare the effect of cannibalism on the dynamic behavior for the ODE system, the semilinear reaction-diffusion system, and the quasi-linear cross-diffusion system.
In this paper, we first prove that stability properties of equilibria for the ODE model (3) and the semilinear reaction-diffusion model (5) is similar, but under suitable conditions on reaction coefficients, cross-diffusion-driven Turing instability in the quasi-linear model (6) occurs. It is found that the cross diffusion rate α 13 is a decisive factor of destabilizing positive steady state, that is, cannibalism has no longer a stabilizing effect. en, the uniform boundedness and the global existence of time-varying solutions for the cross-diffusion system (6) are investigated by using the energy estimates and Gagliardo-Nirenberg-type inequalities when the space dimension is one. Finally, some criteria on the global asymptotic stability of the positive equilibrium point for (6) are given by Lyapunov function. e obtained results indicate that, for any cannibalism rate e, under certain conditions on the other reaction coefficients, the model has no nonconstant positive steady state if the diffusion matrix is positive definite and the self-diffusion coefficients are large enough.
Moreover, we, in [14], discussed another cross-diffusion model: where d 4 is a cross-diffusion coefficient and ε is a positive constant (please refer to [12]). We proved that if a, b, g, s, r, e, ε, d 1 , d 2 , and d 3 are fixed such that 0 < ε < ε 0 , ag < 1, r > s and (9) (see Section 2) hold, then there exists a positive constant d * 4 such that (7) has at least one nonconstant positive steady state for d 4 ≥ d * 4 . is implies that the cross-diffusion rate is the decisive factor of destabilization on the constant positive steady state and therefore cannibalism is an auxiliary destabilizing force. Recently, in another paper, we detailedly describe the local structure of the nonconstant steady states and discuss the stability and instability of the steady state bifurcation.

ODE Model
, then a positive equilibrium point is given by If r ≠ 0 and br + gs − gr > 0, then a positive equilibrium point is uniquely given by In fact, the stability of these nonnegative equilibria in (3) has been obtained in [1] as follows: (1) E 0 is unconditionally unstable.

Theorem 1.
e positive equilibrium u of (3) is locally asymptotically stable if ag < 1.

Proof.
e Jacobian matrix of (3) to the generic u s � (u s , v s , w s ) reads J u s � j 11 j 12 j 13 where j 11 � − 1 + av s + w s , A direct calculation yields that the characteristic equa- with It is easy to see that α 3 > 0 and α 1 α 2 − α 3 > 0, thanks to ag < 1. erefore, by Routh-Hurwitz criterion, we know that (u, v, w) is locally asymptotically stable.
Notice that the unstable equilibrium points of (3) are also unstable for (5) and (6). erefore, for systems (5) and (6), we only discuss the stability of the equilibrium points which are stable for (3). (5). Let 0 � μ 1 < μ 2 < μ 3 < · · · be the eigenvalues of the operator − Δ on Ω with the homogeneous Neumann boundary condition, and let E(μ i ) be the eigenspace corresponding to

Weakly Coupled Reaction-Diffusion System
For system (5), by the linearization analysis and some similar arguments to the proof of eorem 2 and eorem 3 in [12], we can obtain the following two theorems.

□
In order to obtain the global asymptotic behavior of the solutions to (5) or (6), we need the following result which can be found in [12].

Lemma 1. Let a and b be positive constants. Assume that
Proof. Let (u, v, w) be the unique positive solution of (5).
(1) Define where λ � (e + as)/eg and ρ � 1/e. Obviously, Complexity 5 From eorem 2, Lemma 1, and (20), we know that It follows from the Poincaré inequality that (24) and (25), we have From the first equation of (5), we have Let m ⟶ ∞ in the above equation, and from (25) On the contrary, (20) implies that there exists a subsequence, still denoted by t m , and nonnegative Combining this with (25)-(28), one can obtain that e global asymptotic stability of E 3 follows from this together with eorem 2.

Remark 1.
e stability of u is demonstrated specifically in [14], that is, u is locally asymptotically stable if ag < 1.

Strongly Coupled Cross-Diffusion System (6).
Comparing Sections 2.1 and 2.2, we find that E i , i � 1, 2, 3, and u have the same stability properties in systems (3) and (5). For the sake of convenience, we denote nonnegative equilibria E i , i � 1, 2, 3, and u by u s � (u s , v s , w s ). Now, we 6 Complexity show that the destabilization effect of cross-diffusion on u s � (u s , v s , w s ).
Linearizing system (6) at an equilibrium (u s , v s , w s ), we can obtain , and J is given in (11).
We denote then the corresponding characteristic polynomial of L is where with where M ij and M ij , i, j � 1, 2, 3, are cofactors of matrix P and J, respectively. erefore, according to the principle of the linearized stability ((21], 8.6), (22], 5.2)), the local stability of nonnegative equilibria of model (6) is given below. en, the following statements for system (6) hold.
(1) u s is locally asymptotically stable if and only if for every i ∈ N, all the eigenvalues of the linearization matrix L have negative real part.
(2) u s is unstable if and only if there exists an i ∈ N, such that the linearization matrix L has at least one eigenvalue with positive real part.
By applying the Routh-Hurwitz criterion or Corollary 2.2 in [23], we have the following stability and instability results.
On the contrary, the constant equilibrium is unstable if and only if there exists an i ∈ N such that c 3 < 0 or c 1 c 2 < c 3 .
anks to calculate c 1 c 2 − c 3 is complicated, we now only discuss the case of c 3 < 0.
To demonstrate the results of stability and instability for u, we give the following examples. (1) Let 8 Complexity r � 4,  us, u is unstable due to Corollary 1 (6).

The Uniform Boundedness and the Global Existence of Solutions for (6)
In this section, the uniform boundedness and the global existence of solutions for (6) are investigated when the space dimension is one. For simplicity, let Ω � (0, 1), denote | · | k,p � ‖ · ‖ W k p (0,1) , | · | p � ‖ · ‖ L p (0,1) , and Q T � Ω × (0, T). From the consequence of a series of important papers [24][25][26] where T ≤ + ∞ is the maximal existence time for the local solution. If the solution (u, v, w) satisfies the estimates 1)). In order to establish the timing uniform W 1 2 − estimate of the solution for system (6), the following corollaries to the Gagliardo-Nirenberg-type inequality play important roles.

Complexity
In this section, we always denote that C is Sobolev embedding constant or other kind of universal constant, A j , B j , and C j are some positive constants which depend only on α ij (i, j � 1, 2, 3), a, b, g, s, r, e, and K j are positive constants depending on d i and α ij (i, j � 1, 2, 3),  a, b, g, s, r, e. When d 1 , d 2 and d but do not on d 1 , d 2 , and d 3 .
Proof. By the maximum principle, one can obtain that the solutions of (6) with nonnegative initial values are always nonnegative. We will give the W 1 2 -estimates of the solution (u, v, w) for (6) next.

Complexity 11
where y � 1 0 (P 2 x + Q 2 x + R 2 x )dx, q is given by (58). It is not hard to verify by (H2) that there exists a positive constant C 4 depending only on α ij (i, j � 1, 2, 3) such that erefore, Applying the Young inequality, the Hӧlder inequality, the Gagliardo-Nirenberg inequality, and estimate (68), one can obtain the following estimates for the terms on the right-hand side of (71): 12 Complexity , us, Complexity erefore,