Structure and Stability of Steady State Bifurcation in a Cannibalism Model with Cross-Diffusion

This paper deals with spatial patterns of a predator-prey crossdiﬀusion model with cannibalism. By applying the asymptotic analysis and Rabinowitz bifurcation theorem, we consider the local structure of steady state to the model and determine an explicit formula of the nonconstant steady state. Furthermore, the criteria of the stability/instability for the steady state with small amplitude are established.


Introduction
Multicomponent system is widely existed in nature and engineering apllications, for instance, in combustion, chemical reactors, tumor growth, gas mixtures, and animal crowds. On the diffusive level, these systems can be described by crossdiffusion equations taking into account multicomponent diffusion and reaction [1]. Specifically, crossdiffusion is a phenomenon in which the concentration gradient of one species induces a flux of the other species. e possibility of crossdiffusion terms in multicomponent systems was proposed by Onsager and Fuoss [2], while Baldwin et al. [3] undertook the experimental verification of the existence of crossdiffusion and also observed that the crossdiffusion coefficients can be quite significant. Since then, various crossdiffusion mathematical models have been suggested to interpret and predict many interesting features of natural multicomponent dynamics [4][5][6][7][8][9][10].
For example, Darcy's law implies that the velocity is proportional to the negative pressure gradient, and the pressure is defined by a state equation imposed by the volume extension of the mixture. Druet and Jungel [9] considered the convective transport in a multicomponent isothermal compressible fluid subject to the mass continuity equations. According to the idea that decomposes the system into a porous-medium-type equation for the volume extension and transport equations for the modified number fractions, they proved the global-in-time existence of classical and weak solutions in a bounded domain with nopenetration boundary conditions. Jungel and Ptashnyk [10] considered crossdiffusion systems defined in a heterogeneous medium, where the heterogeneity is reflected in spatially periodic diffusion coefficients or by the perforated domain. By combining two-scale convergence and the boundedness-by-entropy method, they proved two-scale homogenization limits of parabolic crossdiffusion systems in a heterogeneous medium with no-flux boundary conditions. Practical applications of such approach would investigate the problem of reducing a heterogeneous material to a homogeneous, specifically, predicting the response of refractory concrete to high-temperature exposure in steel furnaces. In nature, the impact of heat and mass transfer processes become particularly evident at high temperatures, where the increased pressure in pores, large temperature gradients, and temperature-induced creep may lead to catastrophic service failures [11]. Benes et al. [12] discussed a nonlinear numerical scheme arising from the implicit time discretization of the Bazant-onguthai model (with crossdiffusion) for hygrothermal behavior of concrete at high temperatures. By theoretical analysis and numerical simulation, they found that the model reproduces well the rapid increase of pore pressure in wet concrete due to extreme heating. In particular, there are three different zones which can be predicted: one corresponds to elements failed due to spalling damage, the other one indicates the part of the structure in which the local strength is sufficiently high to sustain the pore pressure, but its stability is lost due the explosive spalling of the former region, and the last one shows the portion of the cross-section is still capable of transmitting stresses due to mechanical loading, which is thereby responsible for the structural safety during fire. is phenomenon is particularly meaningful for the safety assessment of concrete structures prone to thermally induced spalling.
In fact, the rapid growth of the field of system biology has further contributed to interest in reaction-diffusion systems. Nowadays, crossdiffusion terms, which are inspired by the model, proposed by Shigesada et al. [13] in 1979 when they have studied the segregation of competing species, also have attracted wide attentions used in reaction-diffusion equations encountered in models from mathematical biology [14][15][16][17][18][19][20][21]. 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 and crossdiffusion. In this paper, we investigate a predator-prey system with both cannibalism and crossdiffusion in the form where u(t), v(t), and w(t) are the biomass of an adult predator, a juvenile predator, and prey at time t. a, b, g, s, r, e, and ε are positive constants. a denotes the cannibalism rate. e positive constants d 1 , d 2 , and d 3 are diffusion coefficients, and d 4 is the crossdiffusion coefficient. e initial values u(x, 0), v(x, 0), and w(x, 0) are nonnegative smooth functions which are not identically zero. Ω ⊂ R N is a bounded domain with smooth boundary zΩ, and v is the outward unit normal vector of the boundary zΩ. e homogeneous Neumann boundary condition indicates that this system is self-contained with zero population flux across the boundary. For more details on the backgrounds of crossdiffusion models, one can see [21,22]. e corresponding kinetic system of (1) was introduced by Magnsson in [23]. In this ODE system, Magnusson found that cannibalism, which is used as a means of population control to prevent them from overbreeding [21][22][23][24][25][26][27][28][29][30][31], has a destabilizing effect. When the large prey carries capacity, the juvenile mortality rate is high and adult recruitment rate is low, and stability of the equilibrium will be lost through Hopf bifurcation, the appearance of which is caused by the increasing level of cannibalism. And sustained oscillations can set in for sufficiently high levels of cannibalism. After, Kaewmanee and Tang [32] optimized the results of Magnusson, they obtained that stability of the equilibrium is lost due to the increase of the cannibalism attack rate past a bifurcation point that depends on other parameters. Later, the stability, topological properties, and types of bifurcations of the ODE system have been studied more explicitly by Marlk and Pribylova [33]. e authors proved that both subcritical and supercritical bifurcations may take place and hence the limits' cycle enclosing the stationary point does not need to be stable. Very recently, Zhang et al. [28] analyzed the effect of cannibalism and illustrated that the cannibalization rate can cause the local stability of the equilibrium changes from stable to unstable to stable, or from unstable to stable to unstable. Moreover, the positive equilibrium must be globally asymptotically stable if the cannibalization rate is large enough. ey also obtained that supercritical/subcritical Hopf bifurcation can occur with the rates of two factors (the cannibalization and the benefit from cannibalism) as bifurcation parameters.
Recently, Fu and Yang [34] considered the corresponding pure diffusion system of (1) (d 4 � 0) and proved that the positive equilibrium in this system has the same stability properties when it is regarded as equilibrium of the ODE system. It is shown that the decisive factor of destabilization for semilinear reaction-diffusion system is still cannibalism. erefore, they further discussed the following strongly coupled crossdiffusion system (1).
Obviously, if r ≠ 0 and br + gs − gr > 0, then (1) has the unique positive equilibrium point and α � abr + s− r − eg, β � br + gs − gr. By the linearization analysis, Fu and Yang proved that positive equilibrium can undergo stability switch from stable in the ODE system and semilinear system to unstable in the crossdiffusion system if d 4 is sufficiently large. is means that the decisive factor of destabilization for the positive equilibrium point in (1) is the crossdiffusion rate, and cannibalism is an auxiliary destabilizing force. Besides, by using the Leray-Schauder degree theory, they also obtained the existence of nonconstant positive steady states. For completeness, we introduce this existence result, which can be proved by similar arguments as eorem 5 in [21].
Theorem 1 (see [34]). Let a, b, g, s, r, e, ε, d 1 , d 2 , and d 3 be fixed positive constants such that 0 < ε < ε 0 , ag < 1, r > s, and (A1) hold. If μ ∈ (μ n , μ n+1 )(n ≥ 2) and the sum Our main purpose in this paper is to describe in detail the local structure of the nonconstant steady states and discuss the stability and instability of bifurcation steady states. e rest of the paper is organized as follows. In Section 2, the local bifurcation analysis is performed to examine the structure of bifurcating steady states. Furthermore, the stability and instability of bifurcating steady states with small amplitude will be given in Section 3; meanwhile, the paper ends with a brief discussion.

Local Structure and Formula of Steady
State Bifurcation e existence of nonconstant steady state of system (1) is established under the condition that μ ∈ (μ n , μ n+1 )(n ≥ 2) and the sum n i�2 dimE(μ i ) is odd in [34]. In this section, using the asymptotic analysis and bifurcation theory similar to those in [35][36][37][38], we choose d 4 as a bifurcation parameter and fix the rest of the parameter to explore the local structure of nonconstant steady states of (1) bifurcating from the constant steady state u in one dimension.
Before proceeding, we present some properties about the negative Laplace operator. 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 μ i in H 1 (Ω). Let X be the closure of [C 1 (Ω)] 3 in [H 1 (Ω)] 3 , ϕ ij : j � 1, 2, . . . , dimE(μ i )} be an orthonormal basis of E(μ i ), and X ij � cϕ ij : c ∈ R 3 . en, In particular, for Ω � (0, l), it is well known that the problem has a sequence of simple eigenvalues whose corresponding eigenfunctions are given by is set of eigenfunctions is an orthogonal basis in L 2 (0, l). For later use, we now define a Banach space X by equipped with usual C 2 norm, and a Hilbert space Y Y � L 2 (0, l) × L 2 (0, l) × L 2 (0, l), (8) with the inner product For the sake of simplify, we investigate the structure of nonconstant positive steady state of (1) in one-dimensional interval Ω � (0, l), i.e., consider the associated elliptic problem: Define the map F: en, u � (u, v, w) is a solution of (10), equivalent to it is a zero-point of the map F. Clearly, Notice that Assume For nonconstant solution u s � (u s , v s , w s ) of (10) bifurcating from u with small amplitude, let Mathematical Problems in Engineering where 0 < τ ≪ 1. Substituting (14) and (15) into (10) and equating the O(τ) and O(τ 2 ) terms, respectively, we derive two systems It is easy to get nonzero solutions for (unique up to a constant multiple for any given positive integer j, and this constant can be absorbed into τ in (15)) for (17) as as long as d 0 4 is given by It is clearly known that d 0 Here, in order to get the uniqueness of solution, we need assume that d 0j 4 ≠ d 0k 4 for any integer j ≠ k. Set for a positive integer j 0 . Since d 4 is regarded as the bifurcation parameter, d 0j 4 , j � 1, 2, . . ., is called the possible bifurcation location for the formation of new patterns [38].
is means that the first bifurcation occurs when the parameter d 4 crosses the bifurcation value d 4 min . If the bifurcation is stable, it will be the pattern (u s , v s , w s ) with formulae given in (15) and (20).
In the following, we find the formula of d 1 4 . e adjoint system of the homogeneous system associated with (18) is which has one solution

Stability of Steady State Bifurcation
is section is devoted to study the stability of the pattern solution (u sj , v sj , w sj ) bifurcated from (d 0j 4 , u) by analyzing the sign of the principal eigenvalue.