Dynamics of a Diffusive Predator-Prey Model with Modified Leslie-Gower Term and Michaelis-Menten Type Prey Harvesting

Abstract

A diffusive predator-prey model with modified Leslie-Gower term and Michaelis-Menten type prey harvesting subject to the homogeneous Neumann boundary condition is considered. We obtain the local and global stability of constant equilibria by eigenvalue analysis and iteration technique. Choosing some parameter concerning with harvesting as Hopf bifurcation parameter, we conclude the existence of periodic solutions near positive constant equilibrium. Using the normal form and center manifold theory, and numerical simulations, we demonstrate our theoretical results of stability and direction of periodic solutions. We also derive the non-existence and existence of non-constant positive steady states by energy method and degree theory.

Appendix:  Direction of Hopf Bifurcation

In this appendix, we compute $\operatorname {Re}c_{1}(h_{H}^{j})$. We adopt the same notations of [25]. When $h=h_{H}^{j}$, we set

$$\begin{aligned} q&=\cos\frac{jx}{l}(a_j,b_j)^{\mathrm{T}}=\cos \frac{jx}{l} \biggl(1,- \biggl(\rho+\frac{d_2 j^2}{l^2} \biggr) \frac{1}{\tilde{B}}+\frac {\omega_0}{\tilde{B}}i \biggr)^{\mathrm{T}}, \\ q^*&=\cos\frac{jx}{l}\bigl(a_j^*,b_j^* \bigr)^{\mathrm{T}}=\cos\frac {jx}{l} \biggl(\frac{1}{l\pi}+ \frac{1}{\omega_0 l\pi} \biggl(\frac {d_2j^2}{l^2}+\rho\biggr)i,\frac{\tilde{B}}{l\pi\omega_0}i \biggr)^{\mathrm{T}}, \end{aligned}$$

where

$$\omega_0=\sqrt{ \bigl(-{d^2_2j^4}/{l^4}-{2d_2j^2 \rho}/{l^2}-\rho(\rho+{\tilde{B}}/{\beta} ) \bigr)}. $$

When j≠0, we get

$$g_{20}=\bigl\langle q^*,Q_{qq}\bigr\rangle =0,\qquad g_{11}=\bigl\langle q^*,Q_{q\bar{q}}\bigr\rangle =0,\qquad g_{02}= \bigl\langle q^*,Q_{\bar{q}\bar{q}}\bigr\rangle =0. $$

So it remains to calculate

$$g_{21}=2\bigl\langle q^*,Q_{w_{11}q}\bigr\rangle +\bigl\langle q^*,Q_{w_{20}\bar{q}}\bigr\rangle +\bigl\langle q^*,C_{qq\bar{q}}\bigr\rangle . $$

It is straightforward to compute that

$$\begin{aligned} c_j&=f_{uu}+2f_{uv}b_j,\qquad d_j=g_{uu}+2g_{uv}b_j+g_{vv}b_j^2, \\ e_j&=f_{uu}+f_{uv}(\overline{ b_j}+b_j),\qquad f_j=g_{uu}+g_{uv}( \overline{b_j}+b_j)+g_{vv}|b_j|^2, \\ g_j&=f_{uuu}+f_{uuv}(2b_j+\overline{ b_j}),\qquad h_j=g_{uuu}+g_{uuv}(2b_j+ \overline{b_j})+g_{uvv}\bigl(2|b_j|^2+b_j^2 \bigr), \end{aligned}$$

with

$$\begin{aligned} \left\{ \begin{array}{@{}rcl@{}} f_{uu}&=&-2+\frac{2\alpha m }{\beta(m+\hat{u}_1(\tilde{h}) )^2}+\frac{2c\tilde{h}}{ (c+\hat{u}_1(\tilde{h}) )^3},\qquad f_{uv}=-\frac{\alpha m}{ (m+\hat{u}_1(\tilde{h}) )^2},\\ g_{uu}&=&-\frac{2\rho}{\beta(m+\hat{u}_1(\tilde{h}) )},\qquad g_{uv}=\frac{2\rho}{m+\hat{u}_1(\tilde{h})},\qquad g_{vv}=-\frac {2\rho\beta}{m+\hat{u}_1(\tilde{h})},\\ f_{uuu}&=&-\frac{6\alpha m }{\beta(m+\hat{u}_1(\tilde{h}) )^3}-\frac{6c\tilde{h}}{ (c+\hat{u}_1(\tilde{h}) )^4},\qquad f_{uuv}=\frac{2\alpha m}{ (m+\hat{u}_1(\tilde{h}) )^2},\\ g_{uuu}&=&\frac{6\rho}{\beta(m+\hat{u}_1(\tilde{h}) )^2},\qquad g_{uuv}=-\frac{4\rho}{ (m+\hat{u}_1(\tilde{h}) )^2},\\ g_{uvv}&=&\frac{2\rho\beta}{ (m+\hat{u}_1(\tilde{h}) )^2}, \qquad f_{vv}=f_{uvv}=f_{vvv}=g_{vvv}=0. \end{array} \right. \end{aligned}$$

From computations, it follows that

$$\begin{aligned} H_{20}& \begin{aligned}[t] &=Q_{qq}-\bigl\langle q^*,Q_{qq}\bigr\rangle q-\bigl\langle \overline{q^*},Q_{qq}\bigr\rangle \overline{q} \\ &= \begin{cases}{ -G_3-{2\omega_0f_{uv}i}/{\tilde{B}}\choose {G_2\omega_0}/{\tilde{B}}+{G_3\tilde{A}}/{\tilde{B}}+{(2G_2-G_1)\omega _0i}/{\tilde{B}}} ,&j=0,\\ {c_j\choose d_j} \cos^2\frac{j}{l}x,&j\neq0, \end{cases} \end{aligned} \\ H_{11}&=Q_{q\bar{q}}-\bigl\langle q^*,Q_{q\bar{q}}\bigr\rangle q-\bigl\langle \overline{q^*},Q_{q\bar{q}}\bigr\rangle \bar{q} = \begin{cases} { -G_3\choose{\tilde{A}}G_3/{\tilde{B}}},&j=0,\\ {e_j\choose f_j} \cos^2\frac{j}{l}x,&j\neq0, \end{cases} \\ \bigl\langle q^*,Q_{qq}\bigr\rangle &= \begin{cases} f_{uu}+G_1+ ({2\omega_0 f_{uv}}/{\tilde{B}}-G_2 )i, &j=0,\\ 0,&j\neq0, \end{cases} \\ \bigl\langle \overline{q^*},Q_{qq}\bigr\rangle &= \begin{cases}f_{uu}-G_1-4{\tilde{A}}f_{uv}/{\tilde{B}}+ ({2\omega_0 f_{uv}}/{\tilde{B}}+G_2 )i, &j=0,\\ 0,&j\neq0, \end{cases} \\ \bigl\langle q^*,Q_{q\bar{q}}\bigr\rangle &= \begin{cases}G_3-G_2 i,&j=0,\\ 0,&j\neq0, \end{cases} \\ \bigl\langle \overline{q^*},Q_{q\bar{q}}\bigr\rangle &= \begin{cases}G_3+G_2 i,&j=0,\\ 0,&j\neq0, \end{cases} \\ [2i\omega_0 I-\tilde{L}_{2j}]^{-1}&=( \alpha_1+i\alpha_2)^{-1} \begin{pmatrix} 2\omega_0i+{4d_2 j^2}/{l^2}+\rho&\tilde{B} \\ {\rho}/{\beta}& 2\omega_0i-{(d_2-3d_1)j^2}/{l^2}-\rho \end{pmatrix} , \\ [2i\omega_0 I-\tilde{L}_{0}]^{-1}&=( \alpha_3+i\alpha_4)^{-1} \begin{pmatrix} 2\omega_0i+\rho& \tilde{B} \\ {\rho}/{\beta}& 2\omega_0i-{(d_2+d_1)j^2}/{l^2}-\rho \end{pmatrix} , \\ {\tilde{L}_{2j}}^{-1}&=\frac{1}{\alpha_5} \begin{pmatrix} -{4d_2j^2}/{l^2}-\rho&-\tilde{B} \\ -{\rho}/{\beta}& {(d_2-3d_1)j^2}/{l^2}+\rho \end{pmatrix} ,\\ {\tilde{L}_{0}}^{-1}&= \frac{1}{\alpha_6} \begin{pmatrix}-\rho&-\tilde{B} \\ -{\rho}/{\beta}&{(d_2+d_1)j^2}/{l^2}+\rho \end{pmatrix} , \end{aligned}$$

where

$$\begin{aligned} \alpha_1&=-4\omega_0^2- \biggl( \frac{(d_2-3d_1)j^2}{l^2}+\rho\biggr) \biggl(\frac {4d_2j^2}{l^2}+\rho\biggr)- \frac{\rho\tilde{B}}{\beta },\qquad\alpha_2=\frac{6\omega_0j^2(d_1+d_2)}{l^2}, \\ \alpha_3&=-4\omega_0^2- \biggl( \frac{(d_2+d_1)j^2}{l^2}+\rho\biggr)\rho-\frac{\rho\tilde{B}}{\beta},\qquad \alpha_4=- \frac{2\omega_0j^2(d_1+d_2)}{l^2}, \\ \alpha_5&=\frac{(12d_1d_2-4d_2^2)j^4}{l^4}-\frac{(5d_2-3d_1)j^2\rho }{l^2}- \rho^2-\frac{\rho\tilde{B}}{\beta},\\ \alpha_6&=- \frac{\rho j^2(d_1+d_2)}{l^2}-\rho^2-\frac{\rho\tilde{B}}{\beta}, \\ G_1&=2g_{uv}-\frac{2\tilde{A} g_{vv}}{\tilde{B}}, \\ G_2&=\frac{f_{uu}\tilde{A}}{\omega_0}-\frac{2{\tilde{A}}^2f_{uv}}{\tilde{B}\omega_0}+\frac{\tilde{B} g_{uu}}{\omega_0}- \frac {2\tilde{A} g_{uv}}{\omega_0}+ \frac{{\tilde{A}}^2g_{vv}}{\tilde{B}\omega_0}-\frac{\omega _0g_{vv}}{\tilde{B}}, \\ G_3&=f_{uu}-\frac{2\tilde{A} f_{uv}}{\tilde{B}}. \end{aligned}$$

Then

$$w_{20}= \begin{pmatrix}\tau_1 \\ \tau_2 \end{pmatrix} \cos\frac{2j}{l}x+ \begin{pmatrix} \tau_3 \\ \tau_4 \end{pmatrix} ,\qquad w_{11}= \begin{pmatrix}\sigma_1 \\ \sigma_2 \end{pmatrix} \cos\frac{2j}{l}x+ \begin{pmatrix} \sigma_3 \\ \sigma_4 \end{pmatrix} , $$

where

$$\begin{aligned} \tau_1&=\frac{1}{2}(\alpha_1+i \alpha_2)^{-1} \biggl[ \biggl(2i\omega_0+ \frac{4d_2 j^2}{l^2}+\rho\biggr)c_j+d_j\tilde{B} \biggr], \\ \tau_2&=\frac{1}{2}(\alpha_1+i \alpha_2)^{-1} \biggl[\frac{\rho }{\beta}c_j+d_j \biggl(2i\omega_0- \frac{(d_2-3d_1)j^2}{l^2}-\rho\biggr) \biggr], \\ \tau_3&=\frac{1}{2}(\alpha_1+i \alpha_2)^{-1} \bigl[(2i\omega_0+ \rho)c_j+\tilde{B} d_j \bigr], \\ \tau_4&=\frac{1}{2}(\alpha_1+i \alpha_2)^{-1} \biggl[\frac{\rho }{\beta}c_j+d_j \biggl(2i\omega_0- \frac{(d_2+d_1)j^2}{l^2}-\rho\biggr) \biggr], \\ \sigma_1&=\frac{1}{2\alpha_5} \biggl[ \biggl(\frac{4d_2j^2}{l^2}+ \rho\biggr)e_j+\tilde{B} f_j \biggr], \\ \sigma_2&=\frac{1}{2\alpha_5} \biggl[\frac{\rho}{\beta}e_j- \biggl(\rho+\frac{(d_2-3d_1)j^2}{l^2} \biggr)f_j \biggr], \\ \sigma_3&=\frac{1}{2\alpha_6} (\rho e_j+\tilde{B} f_j ), \\ \sigma_4&=\frac{1}{2\alpha_6} \biggl[\frac{\rho}{\beta}e_j- \biggl(\rho+\frac{(d_2+d_1)j^2}{l^2} \biggr)f_j \biggr]. \end{aligned}$$

Hence,

$$2\operatorname{Re}\bigl(c_{1}\bigl(h_H^j\bigr) \bigr)=\operatorname{Re}(g_{21})=2\operatorname{Re}\bigl\langle q^*,Q_{w_{11}q}\bigr\rangle +\operatorname{Re}\bigl\langle q^*,C_{qq\bar{q}} \bigr\rangle +\operatorname{Re}\bigl\langle q^*,Q_{w_{20}\bar{q}}\bigr\rangle , $$

where

$$\begin{aligned} \bigl\langle q^*,Q_{w_{20}\bar{q}}\bigr\rangle &=\frac{l\pi}{4} \bigl\{ {\bar{a}_j}^* \bigl[f_{uu}(\tau_1+2 \tau_3)+f_{uv}\bigl({\bar{b}}_j( \tau_1+2\tau_3)+(\tau_1+2\tau_3) \bigr) \bigr] \\ &\quad+{\bar{b}_j}^* \bigl[g_{uu}(\tau_1+2 \tau_3)+g_{uv} \bigl({\bar{b}}_j( \tau_1+2\tau_3)+\tau_2+2\tau_4 \bigr)+g_{vv}{\bar{b}}_j(\tau_2+2 \tau_4) \bigr] \bigr\} , \\ \bigl\langle q^*,Q_{w_{11}q}\bigr\rangle &=\frac{l\pi}{4} \bigl\{ {\bar{a}_j}^* \bigl[f_{uu}(\sigma_1+2 \sigma_3)+f_{uv}\bigl({\bar{b}}_j(\sigma _1+2\sigma_3)+(\sigma_1+2 \sigma_3)\bigr) \bigr] \\ &\quad+{\bar{b}_j}^* \bigl[g_{uu}(\sigma_1+2 \sigma_3)+g_{uv}\bigl({\bar{b}}_j( \sigma_1+2\sigma_3)+\sigma_2+2 \sigma_4\bigr)+g_{vv}{\bar{b}}_j( \sigma_2+2\sigma_4) \bigr] \bigr\} , \\ \bigl\langle q^*,C_{qq\bar{q}}\bigr\rangle &=\frac{3l\pi}{8} \bigl({\bar{a}_j}^*g_j+{\bar{b}_j}^*h_j \bigr), \\ \operatorname{Re}\bigl\langle q^*,Q_{w_{20}\bar{q}}\bigr\rangle &=\frac{1}{4} \biggl[f_{uu} \bigl(\tau^R_1+2 \tau^R_{3} \bigr) +f_{uv} \bigl(b^R_j \bigl(\tau_1^R+2\tau^R_3 \bigr)+b^I_j\bigl(\tau^I_1+2 \tau^I_3\bigr) +\bigl(\tau^R_2+2 \tau^R_4\bigr) \bigr) \\ &\quad+ \biggl(\frac{d_2 j^2}{l^2\omega_0}+\frac{\rho}{\omega _0} \biggr) \bigl(f_{uu}\bigl(\tau_1^I+2 \tau_3^I\bigr)+ f_{uv} \bigl( \tau_2^I+2\tau_4^I+ b^R_j\bigl(\tau_1^I+2 \tau_3^I\bigr) \\ &\quad-b^I_j\bigl( \tau_1^R+2\tau_3^R\bigr) \bigr) \bigr) +\frac{\tilde{B}}{\omega_0} \bigl(g_{uu}\bigl(\tau_1^I+2 \tau_3^I\bigr) +g_{uv} \bigl(\bigl( \tau_2^I+2\tau_4^I \bigr)\\ &\quad-b^I_j\bigl(\tau_1^R+2 \tau_3^R\bigr) +b^R_j\bigl( \tau_1^R+2\tau_3^R\bigr) \bigr) \bigr) \\ &\quad+\frac{\tilde{B}}{\omega_0} \bigl(g_{vv}\bigl(b^R_j \bigl(\tau_2^I+2\tau_4^I\bigr) -b^I_j\bigl(\tau_2^R+2 \tau_4^R\bigr)\bigr) \bigr) \biggr], \\ \operatorname{Re}\bigl\langle q^*,Q_{w_{11}q}\bigr\rangle &= \frac{1}{4} \biggl[f_{uu}(\sigma_1+2 \sigma_3)+f_{uv} \bigl( (\sigma_1+2\sigma _3)+b^R_j(\sigma_1+2 \sigma_3) \bigr) \\ &\quad+ \biggl(\frac{d_2 j^2}{l^2\omega_0}+\frac{\rho}{\omega_0} \biggr)g_{uv} b^I_j(\sigma_1+2\sigma_3)\\ &\quad+ \frac{\tilde{B}}{\omega_0} \bigl(g_{uv}b^I_j( \sigma_1+2\sigma_3)+g_{vv}b^I_j (\sigma_2+2\sigma_4) \bigr) \biggr], \\ \operatorname{Re}\bigl\langle q^*,C_{qqq}\bigr\rangle &=\frac{3}{8} \biggl(f_{uuu}-\frac{2(f_{uuv}+f_{uuv})({d_2 j^2}+\rho l^2)}{\tilde{B} l^2}+g_{uuv} \biggr), \end{aligned}$$

where $\xi^{R}=\operatorname{Re} \xi$, and $\xi^{I}=\operatorname{Im} \xi$.

Likewise, when j=0, we derive $\tilde{A}=\rho$, and

$$\begin{aligned} \operatorname{Re}\bigl(c_{1}\bigl(h_H^0\bigr) \bigr)&=\operatorname{Re}\bigl\langle q^*,Q_{w_{11}q}\bigr\rangle + \frac{1}{2}\operatorname{Re}\bigl\langle q^*,C_{qq\bar{q}}\bigr\rangle + \frac{1}{2}\operatorname{Re}\bigl\langle q^*,Q_{w_{20}\bar{q}}\bigr\rangle \\ &\quad+ \operatorname{Re} \biggl(\frac{ i w_0}{2}\bigl\langle q^*,Q_{qq}\bigr\rangle \cdot\bigl\langle q^*,Q_{q\bar{q}}\bigr\rangle \biggr) \\ &=\frac{\beta}{2\rho(\beta\rho+\tilde{B})} \biggl[2(f_{uv}+g_{vv}) \biggl( \frac{\rho}{\beta}+\frac{\rho^2}{\tilde{B}} \biggr)G_3 +(f_{uuu}+g_{uuv}) \biggl(1-\frac{2\rho}{\tilde{B}} \biggr) \biggr] \\ &\quad+\frac{\beta}{6\rho(\beta\rho+\tilde{B})} \biggl(f_{uu}-\frac {\rho}{\tilde{B}}f_{uv}- \frac{\rho}{\tilde{B}}-g_{uv} \biggr) \bigl(-4\omega_0^2G_3+ \omega_0G_2 \bigr)\\ &\quad- \biggl(\frac{2\omega _0f_{uv}}{\tilde{B}}-G_2 \biggr)G_3\omega_0 \\ &\quad+\frac{\beta}{6\rho(\beta\rho+\tilde{B})} \biggl\{ \biggl(\frac{\omega_0}{\tilde{B}}f_{uv}- \frac{\rho^2}{\tilde{B}\omega _0}+\frac{\tilde{B}}{\omega_0}g_{uu} \biggr) \biggl(\omega _0(2G_2-G_1)\\ &\quad-\frac{2\omega_0(4\omega_0^2+\rho)f_{uv}}{\tilde{B}} \biggr) \\ &\quad+f_{uv} \biggl[\frac{4 f_{uv}\omega_0^2}{\tilde{B}} \biggl(1+\frac{\rho}{\beta} \biggr)+G_2 \biggl(\frac{\rho\omega_0}{\beta }+\frac{\rho^2\omega_0}{\beta}- \frac{4\omega_0^3}{\tilde{B}} \biggr) \biggr] \\ &\quad+f_{uv} \biggl[G_3 \biggl(\frac{\rho^2-\rho}{\tilde{B}} \biggr)-\frac{2\omega_0^2}{\tilde{B}} (2G_2-G_1 ) \biggr] \\ &\quad+ \biggl(\frac{\rho}{\omega_0}+g_{uv} \biggr) \biggl(- \frac {4\omega_0\rho G_3}{\beta}-\frac{4\omega_0\rho G_2}{\tilde{B}}+\frac {2\omega_0\rho G_3}{\tilde{B}} \biggr) \biggr\} + \frac{\omega _0G_2(f_{uu}+G_1)}{2}. \end{aligned}$$