The instability of periodic solutions for a population model with cross-di ﬀ usion

: This paper is concerned with a population model with prey refuge and a Holling type III functional response in the presence of self-di ﬀ usion and cross-di ﬀ usion, and its Turing pattern formation problem of Hopf bifurcating periodic solutions was studied. First, we discussed the stability of periodic solutions for the ordinary di ﬀ erential equation model, and derived the ﬁrst derivative formula of periodic functions for the perturbed model. Second, applying the Floquet theory, we gave the conditions of Turing patterns occurring at Hopf bifurcating periodic solutions. Additionally, we determined the range of cross-di ﬀ usion coe ﬃ cients for the di ﬀ usive population model to form Turing patterns at the stable periodic solutions. Finally, our research was summarized and the relevant conclusions were simulated numerically.


Introduction
Since 1946, biologist Crombic proved the stability effect through experiments [1,2] and more and more scholars analyzed the refuge effect on the population model [3][4][5][6][7][8][9], mainly focused on the self-diffusion effect on dynamic behavior of the population system.In addition to the effect of selfdiffusion, cross-diffusion also plays an important role during the population pattern formation.About the predator-prey systems with diffusion terms, many scholars have studied the Turning instability and Hopf bifurcation of its constant equilibrium [10][11][12][13][14][15][16][17].At present, for the reaction-diffusion predatorprey system, most literatures [18][19][20][21][22][23][24][25] focus on Turing instability of the constant equilibrium, but there are few research results on the stability of the periodic solutions.Therefore, it is significant to study the Turing pattern formation of Hopf bifurcating periodic solutions for the cross-diffusion population model with prey refuge and the Holling III functional response.
In 2015, Yang et al. [9] studied a diffusive prey-predator system in Holling type III with a prey refuge: Here, u, v indicates the quantity of prey and predator respectively; α, β, a, r, c, k are all positive; a is the intrinsic growth rate of the prey; a/r represents the maximum carrying capacity of the environment on the prey; c is the mortality rate of the predator; k represents the conversion rate after the predator eating the prey; m ∈ [0, 1) indicates the refuge coefficient, i.e., the proportion of the protected prey.Only (1 − m)u can be caught by the predator.In the real world, the mobility of each species is affected not only by itself but also by the density of other species.Therefore, on the basis of (1.1), we introduce the cross-diffusion terms and establish the population model as follows: where Ω = (0, lπ) is a bounded domain with smooth boundary ∂Ω in R n and D 11 , D 22 and D 12 , D 21 are the self-diffusivity and cross-diffusivity of u and v.We assume that the diffusion coefficients satisfy The organizational structure of the rest is as follows: In section two, we study the stability of Hopf bifurcating periodic solutions for the ordinary differential population model and derive the first derivative formula of the periodic function for the corresponding perturbed model.In section three, we give the conditions of Turing patterns occurring at Hopf bifurcating periodic solutions in the reaction-diffusion population system.In section four, we give a brief conclusion.Finally, the relevant conclusions are verified by numerical simulations.

Dynamics of the zero-dimensional population model
In order to research conveniently, we nondimensionalize model (1.2).Let û = u β , v = v kβ , t = at, and we still replace û, v, t with u, v, t, then model (1.2) becomes where, a , θ = c a and p = rβ a , s = kα a .

Stability of periodic solutions of the ordinary differential population model
The ordinary differential equations corresponding to the reaction-diffusion population model (2.1) are (2.2) By calculation, four equilibria of model (2.2) are P 0 = (0, 0), P 1 = (1/p, 0), Clearly, the equilibrium P − = (u − , v − ) has no biological significance, so we do not study its dynamic behavior.Let's make the following assumptions: 2θp and assume that (A 1 ) satisfies.The following results are true for model (2.2).

Dynamics of the perturbed population model
In this subsection, for the perturbed population model, we derive the first derivative formula of the periodic function about the perturbation coefficients.On the basis of model (2.1), we introduce the perturbation term τ and coefficients The corresponding perturbed population model is where τ is sufficiently small such that 1 + τk 11 τk 12 τk 21 1 + τk 22 is reversible, then (2.7) can be reduced to At P + = (κ, v κ ), the Jacobian matrix of (2.8) is where, (2.10) Let the characteristic equation corresponding to J(κ, τ) be where (2.12) When κ → κ τ , let β(κ τ ) ± i ω(κ τ ) be the roots of the characteristic Eq (2.11), then (2.13) Lemma 2.1.(See [26]) When κ → κ 0 , the population model (2.2) has a stable periodic solution u T (t), v T (t) bifurcating from P + = (κ, v κ ) and T is the minimum positive period of u T (t), v T (t) .Then there is a positive number τ 1 such that for any τ ∈ (−τ 1 , τ 1 ), the perturbed population model (2.7) has a periodic solution u T (t, τ), v T (t, τ) depending on if τ, T (τ) is the minimum positive periodic function.

Turing patterns of periodic solutions for the reaction-diffusion population model
With respect to the population model (1.2) and according to the theory expounded in [27], we study the mathematical mechanisms of Turing patterns occurring at the stable periodic solution u T (t), v T (t) .By the first derivative formula of the periodic function of the perturbed population model (2.7), we give the following theorem.Theorem 3.1.If hypothesis (A 4 ) and Re (c 1 (κ 0 )) < 0 hold, when κ → κ 0 , the stable spatially homogeneous Hopf bifurcating periodic solution bifurcates u T (t), v T (t) from P + = (κ, v κ ).If the domain Ω is large enough and Re (c 1 (κ 0 )) and Im (c 1 (κ 0 )) from (2.5) and (2.6): .
Proof.Let the linearized vector form of the population model (2.1) at u T (t), v T (t) be ∂φ ∂t , ∂ϕ ∂t where, is the Jacobian matrix of model (2.1) at u T (t), v T (t) .D := d 11 d 12 d 21 d 22 , ∆ is the Laplace operator.