Hopf bifurcation analysis in a diffusive predator–prey system with spatial heterogeneity and delays

Abstract

In this paper, we consider a diffusive Lotka–Volterra predator–prey system with spatial heterogeneity and two delays. We first show that there exists a nonconstant positive steady state when the diffusion rates of prey and predator are large. Then, we obtain the stability of the steady state and show the existence of a Hopf bifurcation. Moreover, some numerical simulations are provided to illustrate our theoretical results.

Appendix

In this subsection, we study the stability change of the positive steady state $U_{\lambda }$ when $\tau$ and r are proportional. Without loss of generality, we let $\tau =\delta r$, where $\delta >0$ is a fixed positive constant. The method used is similar to that in Sect. 3. In this case, the infinitesimal generator of the solution semigroup of (3.4) can be defined by

$$\begin{aligned} \mathfrak {A}_{\lambda }(r)\psi =\dot{\psi }, \end{aligned}$$

with

$$\begin{aligned} \mathcal {D}(\mathfrak {A}_{\lambda }(r))=\left\{ \psi \in C_{\mathbb {C}}\cap C_{\mathbb {C}}^{1}:\dot{\psi }(0) =A_{\lambda }\psi (0)-B_{\lambda }\psi (-\delta r)+C_{\lambda }\psi (-r) \right\} , \end{aligned}$$

and $C_{\mathbb {C}}=C([-\max \{\delta r,r\},0],X^{2}_{\mathbb {C}})$, $C_{\mathbb {C}}^{1}=C^{1}([-\max \{\delta r,r\},0],X^{2}_{\mathbb {C}})$. Then, the spectrum of $\mathfrak {A}_{\lambda }(r)$ is

$$\begin{aligned} \sigma (\mathfrak {A}_{\lambda }(r))=\{\mu \in \mathbb {C}:\Pi (\lambda ,\mu ,r)\psi =0~\text {for some}~\psi \in X_{\mathbb {C}}^{2}\setminus \{\textbf{0}\}\}, \end{aligned}$$

where

$$\begin{aligned} \Pi (\lambda ,\mu ,r)\psi =A_{\lambda }\psi -B_{\lambda }\psi e^{-\mu \delta r}+C_{\lambda }\psi e^{-\mu r}-\mu \psi =0. \end{aligned}$$

(4.2)

We first analyze whether $\mathfrak {A}_{\lambda }(r)$ has purely imaginary eigenvalues when $r>0$ and $\bar{a}_{1}\bar{a}_{4}<\bar{a}_{2}\bar{a}_{3}$. Similar to the case where $\tau$ is fixed and r changes, set $\mu =\pm \textrm{i}\omega (\omega >0)$, $\omega =\lambda h$, $\theta =\omega r$, and ignore a scale factor, take $\psi =(\psi _1,\psi _2)^T$, where

$$\begin{aligned} {\left\{ \begin{array}{ll} &{}\psi _{1}=s+\lambda z_{1}, ~\psi _{2}=R_{1}+\textrm{i}R_{2}+\lambda z_{2},\\ &{}\Vert \psi _{1}\Vert _2^2+\Vert \psi _{2}\Vert _2^2=|\Omega |,\\ &{}s\ge 0,~R_{i}\in \mathbb {R},~\text {and}~z_{i}\in (X_{1})_{\mathbb {C}},~i=1,2. \end{array}\right. } \end{aligned}$$

(4.3)

Then, (4.2) becomes

$$\begin{aligned} \begin{aligned} \check{g}_{1}(z_{1},z_{2},s,R_{1},R_{2},h,\theta ,\lambda )&=\Delta z_{1}+[r_{1}(x)-2a_{1}(x)u_{\lambda }-a_{2}(x)v_{\lambda }](s+\lambda z_{1})\\ {}&-a_{2}(x)u_{\lambda }e^{-\text {i}\delta \theta }(R_{1}+\text {i}R_{2}+\lambda z_{2})-\text {i}h(s+\lambda z_{1})=0,\\ \check{g}_{2}(z_{1}, z_{2},s,R_{1},R_{2},h,\theta ,\lambda )&=d\Delta z_{2}+[-r_{2}(x)+a_{3}(x)u_{\lambda }-2a_{4}(x)v_{\lambda }](R_{1}+\text {i}R_{2}+\lambda z_{2})\\ {}&+a_{3}(x)v_{\lambda }e^{-\text {i}\theta }(s+\lambda z_{1})-\text {i}h(R_{1}+\text {i}R_{2}+\lambda z_{2})=0,\\ \check{g}_{3}(z_{1}, z_{2},s,R_{1},R_{2},h,\theta ,\lambda )&=(s^2+R_1^2+R_2^2)|\Omega |-|\Omega |+\lambda ^2(\Vert z_1\Vert _2^2+\Vert z_2\Vert _2^2 )=0. \end{aligned} \end{aligned}$$

Define $\check{G}:(X_{1})_{\mathbb {C}}^{2}\times \mathbb {R}^{6}\mapsto Y_{\mathbb {C}}^{2}\times \mathbb {R}$ as $\check{G}=(\check{g}_{1},\check{g}_{2},\check{g}_{3})$. If

$$\begin{aligned} {\left\{ \begin{array}{ll} \check{G}(z_{1},z_{1},s,R_{1},R_{2},h,\theta ,\lambda )=0,\\ z_{1},z_{2}\in (X_{1})_{\mathbb {C}}, s\ge 0, R_{1},R_{2}\in \mathbb {R}, h>0, \theta \in (0,2\pi ] \end{array}\right. } \end{aligned}$$

(4.4)

is solvable, then $\mathfrak {A}_{\lambda }(r)$ has purely imaginary eigenvalues $\mu =\textrm{i}\omega =\textrm{i} \lambda h$. Using similar arguments as in Lemma 3.2 and Theorem 3.3, we have the following result.

Theorem 4.1

Suppose that (2.2) holds, and $\bar{a}_{1}\bar{a}_{4}<\bar{a}_{2}\bar{a}_{3}$. Then, there exists $0<\check{\lambda }\ll 1$, and a continuously differential mapping $\lambda \mapsto (\check{z}_{1\lambda }, \check{z}_{2\lambda },\check{s}_{\lambda },\check{R}_{1\lambda },\check{R}_{2\lambda }, \check{h}_{\lambda },\check{\theta }_{\lambda })$ from $[0,\check{\lambda }]$ to $(X_{1})^{2}_{\mathbb {C}}\times \mathbb {R}^{5}$, such that (4.4) has a unique solution $(\check{z}_{1\lambda }, \check{z}_{2\lambda },\check{s}_{\lambda },\check{R}_{1\lambda },\check{R}_{2\lambda }, \check{h}_{\lambda },\check{\theta }_{\lambda })$. Moreover, when $\lambda =0$,

$$\begin{aligned} \begin{aligned}&\check{h}_{0}=\left[ \frac{-(\bar{a}_{4}^{2}c_{0}^{2}+\bar{a}_{1}^{2}\beta _{0}^{2})+\sqrt{(\bar{a}_{4}^{2}c_{0}^{2}-\bar{a}_{1}^{2}\beta _{0}^{2})^{2} +4(\bar{a}_{2}\bar{a}_{3}\beta _{0}c_{0})^{2}}}{2|\Omega |^2}\right] ^{1/2},\\&\cos [(1+\delta )\check{\theta }_{0}]=\frac{\check{h}_{0}^{2}|\Omega |^2-\bar{a}_{1}\bar{a}_{4}\beta _{0}c_{0}}{\bar{a}_{2}\bar{a}_{3}\beta _{0}c_{0}}, ~\sin [(1+\delta )\check{\theta }_{0}]=\frac{\check{h}_{0}|\Omega |(\bar{a}_{4}c_{0}+\bar{a}_{1}\beta _{0})}{\bar{a}_{2}\bar{a}_{3}\beta _{0}c_{0}},\\&\check{s}_0=\frac{\bar{a}_2\beta _0}{\sqrt{(\bar{a}_1^2+\bar{a}_2^2)\beta _0^2+\check{h}_0^2|\Omega |^2}} =\sqrt{\frac{\bar{a}_4^2c_0^2+\check{h}_0^2|\Omega |^2}{(\bar{a}_3^2+\bar{a}_4^2)c_0^2+\check{h}_0^2|\Omega |^2}},\\&\check{R}_{10}=\frac{\check{h}_0|\Omega |\sin (\delta \check{\theta }_0)-\bar{a}_1\beta _0\cos (\delta \check{\theta }_0)}{\sqrt{(\bar{a}_1^2+\bar{a}_2^2)\beta _0^2+\check{h}_0^2|\Omega |^2}},~ \check{R}_{20}=\frac{-\left[ h_0|\Omega |\cos (\delta \check{\theta }_0)+\bar{a}_1\beta _0\sin (\delta \check{\theta }_0)\right] }{\sqrt{(\bar{a}_1^2+\bar{a}_2^2)\beta _0^2+\check{h}_0^2|\Omega |^2}}, \end{aligned} \end{aligned}$$

(4.5)

and $(\check{z}_{10},\check{z}_{20})$ uniquely solves the following system

$$\begin{aligned} {\left\{ \begin{array}{ll} \Delta z_{1}=-[r_{1}(x)-2a_{1}(x)\beta _{0}-a_{2}(x)c_{0}-\textrm{i}\check{h}_{0}]s_0+a_{2}(x)\beta _{0}e^{-\textrm{i}\delta \check{\theta }_{0}}(\check{R}_{10}+\textrm{i}\check{R}_{20}),\\ d\Delta z_{2}=-[-r_{2}(x)+a_{3}(x)\beta _{0}-2a_{4}(x)c_{0}-\textrm{i}\check{h}_{0}](\check{R}_{10}+\textrm{i}\check{R}_{20}) -a_{3}(x)c_{0}\check{s}_0e^{-\textrm{i}\check{\theta }_{0}}. \end{array}\right. } \end{aligned}$$

As a consequence of Theorem 4.1, we obtain the existence of purely imaginary eigenvalues of $\mathfrak {A}_{\lambda }(r)$ as follows.

Theorem 4.2

Suppose that (2.2) holds, and $\bar{a}_{1}\bar{a}_{4}<\bar{a}_{2}\bar{a}_{3}$. Then for $\lambda \in (0,\check{\lambda }]$ with $0<\check{\lambda }\ll 1$, $\Pi (\lambda ,\textrm{i}\omega ,r)\psi =0$ with $\omega >0$, $r>0$ and $\psi \in X^2_\mathbb {C}\setminus \{\textbf{0}\}$ if and only if

$$\begin{aligned} \begin{aligned} \omega&=\check{\omega }_{\lambda }=\lambda \check{h}_{\lambda }, ~r=\check{r}_{\lambda ,n}=\frac{\check{\theta }_{\lambda }+2n\pi }{\check{\omega }_{\lambda }},~n=0,1,2,\cdots .\\ ~\psi&=\check{a}\psi _{\lambda }=\check{a}\left( \begin{array}{c} \check{s}_{\lambda }+\lambda \check{z}_{1\lambda } \\ \check{R}_{1\lambda }+\textrm{i}\check{R}_{2\lambda }+\lambda \check{z}_{2\lambda } \\ \end{array} \right) , \end{aligned} \end{aligned}$$

where $\check{a}$ is a nonzero constant. $\check{z}_{i\lambda }$, $\check{s}_{\lambda }$, $\check{R}_{i\lambda }$, $\check{h}_{\lambda }$, $\check{\theta }_{\lambda }$, $i=1,2$ are defined in Theorem 4.1.

Similarly in Theorems 3.7 and 3.8, we can obtain the simplicity of $\textrm{i}\check{\omega }_{\lambda }$ and the transversality condition. From Lemma 3.10, we know that there exists $0<\check{\lambda }\ll 1$ such that when $\bar{a}_{1}\bar{a}_{4}\le \bar{a}_{2}\bar{a}_{3}$ and $r=0$, all eigenvalues of $\mathfrak {A}_{\lambda }(r)$ have negative real parts for $\lambda \in (0,\check{\lambda }]$. Moreover, similar to Lemma 3.9, we can verify that when $\bar{a}_{1}\bar{a}_{4}>\bar{a}_{2}\bar{a}_{3}$, all real parts of eigenvalues of $\mathfrak {A}_{\lambda }(r)$ are negative for any $r\ge 0$ and $\lambda \in (0,\check{\lambda }]$ for sufficiently small $\check{\lambda }$. Thus, we have the similar result to Theorem 3.11.