Dynamical study of a prey–predator model incorporating nonlinear prey refuge and additive Allee effect acting on prey species

Abstract

In this study, we have analyzed a mathematical model on predator–prey interactions incorporating prey refuge and additive Allee effect on the prey species. The various dynamical behaviors of the system have analyzed, considering the prey refuge is proportional to both the prey and predator species with Beddington–DeAngelis functional response. None, single, or two coexistence equilibria can exist at the first quadrant of the phase space considering strong additive Allee effect in the system. The permanence, local stability, saddle-node bifurcation, existence of a stable limit cycle and Hopf bifurcation are examined under some parametric conditions. We have also calculated the first Lyapunov number to define the nature of Hopf bifurcating periodic solution. Moreover, it has established a parameter subset at which the dynamical system may have a cusp point of co dimension 2 (Bogdanov–Takens bifurcation). Finally, we have executed an adequate numerical simulation to authenticate our analytical findings.

Appendices

Appendix

A1. The values of the coefficients A, B, C, D and E used in Eq. (5) are given bellow

$$\begin{aligned} A= & {} -er\delta \, \left( em-bd \right) ,\\ B= & {} \left( \left( K-h \right) \left( em-bd \right) \delta -cd \right) er,\\ C= & {} -bdehKr\delta +bdeKnr\delta +{e}^{2}hKmr\delta -{e}^{2}Kmnr\delta \\- & {} cdehr+cdeKr+b{d}^{2}K-deKm, \\ D= & {} cdehKr-cdeKnr+b{d}^{2}hK-dehKm+a{d}^{2}K,\\ E= & {} ad^2hK. \end{aligned}$$

A2. The constants $\alpha _{i}$ $(i=1,2,3,4)$ given in local stability section are as follows

$$\begin{aligned}&\alpha _{1}=d \big ( -ce{h}^{2}r{x_{2}^{*}}^{2}-2\,cehr{x_{2}^{*}}^{3}+ceKnr{x_{2}^{*}}^{2}-cer{x_{2}^{*}}^{4}\\&\qquad +bd{h}^{2}Kx_{2}^{*}+2\,bdhK{x_{2}^{*}}^{2}+bdK{x_{2}^{*}}^{3}\\&\qquad +cd{h}^{2}Ky_{2}^{*}+2\,cdhKx_{2}^{*}y_{2}^{*}+cdK{x_{2}^{*}}^{2}y_{2}^{*}-e{h}^{2}Kmx_{2}^{*}\\&\qquad -2\,ehKm{x_{2}^{*}}^{2}-eKm{x_{2}^{*}}^{3} \big ),\\&\alpha _{2}=x_{2}^{*} \big ( -bde{h}^{2}r{x_{2}^{*}}^{2}-2\,bdehr{x_{2}^{*}}^{3}+bdeKnr{x_{2}^{*}}^{2}\\&\qquad -bder{x_{2}^{*}}^{4}+{e}^{2}{h}^{2}mr{x_{2}^{*}}^{2}+2\,{e}^{2}hmr{x_{2}^{*}}^{3}\\&\qquad -{e}^{2}Kmnr{x_{2}^{*}}^{2}+{e}^{2}mr{x_{2}^{*}}^{4}+2\,b{d}^{2}{h}^{2}Ky_{2}^{*}\\&\qquad +4\,b{d}^{2}hKx_{2}^{*}y_{2}^{*}+2\,b{d}^{2}K{x_{2}^{*}}^{2}y_{2}^{*}-2\,de{h}^{2}Kmy_{2}^{*}\\&\qquad -4\,dehKmx_{2}^{*}y_{2}^{*}-2\,deKm{x_{2}^{*}}^{2}y_{2}^{*} \big ),\\&\alpha _{3}=c{d}^{2}e{h}^{2}Ky_{2}^{*}+2\,c{d}^{2}ehKx_{2}^{*}y_{2}^{*}+c{d}^{2}eK{x_{2}^{*}}^{2}y_{2}^{*}\\&\qquad +{e}^{2}{h}^{2}mr{x_{2}^{*}}^{2}+2\,{e}^{2}hmr{x_{2}^{*}}^{3}-{e}^{2}Kmnr{x_{2}^{*}}^{2}\\&\qquad +{e}^{2}mr{x_{2}^{*}}^{4}-b{d}^{2}{h}^{2}Ky_{2}^{*}-2\,b{d}^{2}hKx_{2}^{*}y_{2}^{*}-b{d}^{2}K{x_{2}^{*}}^{2}y_{2}^{*}, \\&\alpha _{4}=y_{2}^{*} \big ( b{d}^{2}e{h}^{2}Kx_{2}^{*}+2\,b{d}^{2}ehK{x_{2}^{*}}^{2}+b{d}^{2}eK{x_{2}^{*}}^{3}\\&\qquad -d{e}^{2}{h}^{2}Kmx_{2}^{*}-2\,d{e}^{2}hKm{x_{2}^{*}}^{2}-d{e}^{2}Km{x_{2}^{*}}^{3}\\&\qquad +{e}^{2}{h}^{2}mr{x_{2}^{*}}^{2}+2\,{e}^{2}hmr{x_{2}^{*}}^{3}-{e}^{2}Kmnr{x_{2}^{*}}^{2}\\&\qquad +{e}^{2}mr{x_{2}^{*}}^{4}-b{d}^{2}{h}^{2}Ky_{2}^{*}-2\,b{d}^{2}hKx_{2}^{*}y_{2}^{*}\\&\qquad -b{d}^{2}K{x_{2}^{*}}^{2}y_{2}^{*} \big ). \end{aligned}$$

A3. The Jacobian matrix of the system (1) at $(x^{*},y^{*})$ is as follows

$$\begin{aligned} J^{*}= \left[ \begin{array}{cc} A_{11} &{} A_{12}\\ A_{21} &{} A_{22}\end{array} \right] , \end{aligned}$$

where

$$\begin{aligned}&A_{11}= rx^{*} \left( -\frac{1}{K}+{\frac{n}{ \left( x^{*}+h \right) ^{2}}} \right) +{\frac{{d}^{2}y^{*}b}{{e}^{2}mx^{*}}},\\&A_{12}={\frac{\delta \,y^{*}d}{e \left( -\delta \,y^{*}+1 \right) }}-{\frac{d}{e}}+{\frac{{d}^{2}y^{*} \left( -b\delta \,x^{*}+c \right) }{{e}^{2}mx^{*} \left( -\delta \,y^{*}+1 \right) }},\\&A_{21}=\left( {\frac{d}{x^{*}}}-{\frac{{d}^{2}b}{emx^{*}}} \right) y^{*},\\&A_{22}=\left( -{\frac{\delta \,d}{-\delta \,y^{*}+1}}-{\frac{{d}^{2} \left( -b\delta \,x^{*}+c \right) }{emx^{*} \left( -\delta \,y^{*}+1 \right) }} \right) y^{*}. \end{aligned}$$

A4 The expressions of $D^{2}F(E^{*},\delta _{sn})(\chi ,\chi )$ and $D^{3}F(E^{*},\delta _{sn})(\chi ,\chi ,\chi )$ are defined by the following quantities

$$\begin{aligned}&D^{2}F(E^{*},\delta _{sn})(\chi ,\chi )\\&\quad =\left[ \begin{array}{c}{\frac{\partial ^2 F_{1}}{\partial x^{2}}\chi _{1}^{2}+\frac{\partial ^2 F_{1}}{\partial x \partial y}\chi _{1}\chi _{2}+\frac{\partial ^2 F_{1}}{\partial y \partial x}\chi _{2}\chi _{1}+\frac{\partial ^2 F_{1}}{\partial y^{2}}\chi _{2}^{2}}\\ {\frac{\partial ^2 F_{2}}{\partial x^{2}}\chi _{1}^{2}+\frac{\partial ^2 F_{2}}{\partial x \partial y}\chi _{1}\chi _{2}+\frac{\partial ^2 F_{2}}{\partial y \partial x}\chi _{2}\chi _{1}+\frac{\partial ^2 F_{2}}{\partial y^{2}}\chi _{2}^{2}}\end{array} \right] ,\\&D^{3}F(E^{*},\delta _{sn})(\chi ,\chi ,\chi )\\&\quad =\left[ \begin{array}{c}{\frac{\partial ^3 F_{1}}{\partial x^{3}}\chi _{1}^{3}+\frac{\partial ^3 F_{1}}{\partial x^{2} \partial y}\chi _{1}^{2}\chi _{2}+\frac{\partial ^3 F_{1}}{\partial y^{2} \partial x}\chi _{2}^{2}\chi _{1}+\frac{\partial ^3 F_{1}}{\partial y^{3}}\chi _{2}^{3}}\\ {\frac{\partial ^3 F_{2}}{\partial x^{3}}\chi _{1}^{3}+\frac{\partial ^3 F_{2}}{\partial x^{2} \partial y}\chi _{1}^{2}\chi _{2}+\frac{\partial ^3 F_{2}}{\partial y^{2} \partial x}\chi _{2}^{2}\chi _{1}+\frac{\partial ^3 F_{2}}{\partial y^{3}}\chi _{2}^{3}}\end{array} \right] . \end{aligned}$$

A5. The expressions of $l_{ij}$ and $m_{ij}$ $(i,j=0,1,2,3)$, shown in  "Behavior of Hopf bifurcation periodic solution" section are as follows

$$\begin{aligned}&l_{10}=r \left( 1-{\frac{x^{*}}{K}}-{\frac{n}{x^{*}+h}} \right) +rx^{*} \left( -{K}^{-1 }+{\frac{n}{ \left( x^{*}+h \right) ^{2}}} \right) \\&\qquad +{\frac{m \left( \delta _{hb}\,y^{*}-1 \right) y^{*} \left( cy^{*}+a \right) }{ \left( b \left( -\delta _{hb}\,y^{*}+1 \right) x^{*}+cy^{*}+a \right) ^{2}}} ,\\&l_{01}=\frac{m \left( -b{\delta _{hb}}^{2}x^{*}{y^{*}}^{2}+2\,b\delta _{hb}\,x^{*}y^{*}+c{y^{*}}^{2}\delta _{hb}+ 2\,ay^{*}\delta _{hb}-bx^{*}-a \right) x^{*}}{ \left( -b\delta _{hb}\,x^{*}y^{*}+bx^{*}+cy^{*}+a \right) ^{2}}, \\&l_{20}={\frac{r \left( - \left( x^{*}+h \right) ^{2}+Kn \right) }{K \left( x^{*}+ h \right) ^{2}}}-{\frac{rx^{*}n}{ \left( x^{*}+h \right) ^{3}}}\\&+{\frac{m \left( \delta _{hb}\,y^{*}-1 \right) ^{2}y^{*}b \left( cy^{*}+a \right) }{ \left( b \left( -\delta _{hb}\,y^{*}+1 \right) x^{*}+cy^{*}+a \right) ^{3}}} ,\\&l_{11}={\frac{m \left( -b{\delta _{hb}}^{2}x^{*}{y^{*}}^{2}+b\delta _{hb}\,x^{*}y^{*}+c{y^{*}}^{2}\delta _{hb}+2\, ay^{*}\delta _{hb}-a \right) }{ \left( -b\delta _{hb}\,x^{*}y^{*}+bx^{*}+cy^{*}+a \right) ^{2}}}\\&\qquad +{\frac{m \left( b{\delta _{hb}}^{2}x^{*}y^{*}+ \left( -bx^{*}-cy^{*}+a \right) \delta _{hb}+2\,c \right) \left( \delta _{hb}\,y^{*}-1 \right) y^{*}bx^{*}}{ \left( -b\delta _{hb}\,x^{*}y^{*}+bx^{*}+cy^{*}+a \right) ^{3}}},\\&l_{02}={\frac{mx^{*} \left( a\delta _{hb}+c \right) \left( bx^{*}+a \right) }{ \left( b \left( -\delta _{hb}\,y^{*}+1 \right) x^{*}+cy^{*}+a \right) ^{3}}},\\&l_{30}=-{\frac{rn}{ \left( x^{*}+h \right) ^{3}}}\\&\qquad +{\frac{rx^{*}n}{ \left( x^{*}+h \right) ^{4}}}+{\frac{m \left( \delta _{hb}\,y^{*}-1 \right) ^{3}y^{*}{b}^{2}}{ \left( \left( -b\delta _{hb}\,x^{*}+c \right) y^{*}+bx^{*}+a \right) ^{3}}}\\&\qquad +{\frac{mx^{*} \left( \delta _{hb}\,y^{*}-1 \right) ^{4}y^{*}{b}^{3}}{ \left( b \left( -\delta _{hb} \,y^{*}+1 \right) x^{*}+cy^{*}+a \right) ^{4}}},\\&l_{21}={\frac{m \left( \delta _{hb}\,y^{*}-1 \right) b \left( 3\,\delta _{hb}\,y^{*}-1 \right) }{ \left( -b\delta _{hb}\,x^{*}y^{*}+bx^{*}+cy^{*}+a \right) ^{2}}}\\&\qquad +{\frac{m \left( \delta _{hb}\,y^{*}-1 \right) ^{2}b \left( 6\,b\delta _{hb}\,x^{*}y^{*}-bx^{*} -2\,cy^{*} \right) }{ \left( -b\delta _{hb}\,x^{*}y^{*}+bx^{*}+cy^{*}+a \right) ^{3}}}\\&\qquad +3{\frac{mx^{*} \left( \delta _{hb}\,y^{*}-1 \right) ^{3}y^{*}{b}^{2} \left( b\delta _{hb}\, x^{*}-c \right) }{ \left( b \left( -\delta _{hb}\,y^{*}+1 \right) x^{*}+cy^{*}+a \right) ^{4 }}},\\&l_{12}={\frac{m \left( {x^{*}}^{2} \left( \delta _{hb}\,y^{*}-1 \right) {b}^{2}+2\,x^{*}y^{*} \left( a\delta _{hb}+c \right) b+a \left( cy^{*}+a \right) \right) \left( a \delta _{hb}+c \right) }{ \left( b \left( -\delta _{hb}\,y^{*}+1 \right) x^{*}+cy^{*}+a \right) ^{4}}},\\&l_{03}={\frac{mx^{*} \left( b\delta _{hb}\,x^{*}-c \right) \left( a\delta _{hb}+c \right) \left( bx^{*}+a \right) }{ \left( b \left( -\delta _{hb}\,y^{*}+1 \right) x^{*}+cy^{*}+a \right) ^{4}}},\\&m_{10}=-{\frac{em \left( \delta _{hb}\,y^{*}-1 \right) \left( cy^{*}+a \right) y^{*}}{ \left( b \left( -\delta _{hb}\,y^{*}+1 \right) x^{*}+cy^{*}+a \right) ^{2}}},\\&m_{01}= -{\frac{emx^{*}y^{*} \left( a\delta _{hb}+c \right) }{ \left( \left( -b\delta _{hb}\,y^{*}+b \right) x^{*}+cy^{*}+a \right) ^{2}}}\\&\qquad +{\frac{em \left( -\delta _{hb}\,x^{*}y^{*}+x^{*} \right) }{a+b \left( -\delta _{hb}\,x^{*}y^{*}+x^{*} \right) +cy^{*}}}-d,\\&m_{20}=-{\frac{em \left( \delta _{hb}\,y^{*}-1 \right) ^{2}b \left( cy^{*}+a \right) y^{*} }{ \left( b \left( -\delta _{hb}\,y^{*}+1 \right) x^{*}+cy^{*}+a \right) ^{3}}},\\&m_{11}=-{\frac{e \left( a\delta _{hb}+c \right) \left( bx^{*} \left( \delta _{hb}\,y^{*}-1 \right) +cy^{*}+a \right) my^{*}}{ \left( b \left( -\delta _{hb}\,y^{*}+1 \right) x^{*}+cy^{*}+a \right) ^{3}}}\\&\qquad -{\frac{e \left( a\delta _{hb}+c \right) \left( bx^{*} \left( \delta _{hb}\,y^{*}-1 \right) +cy^{*}+a \right) m}{ \left( b \left( -\delta _{hb}\,y^{*}+1 \right) x^{*}+cy^{*}+a \right) ^{3}}},\\&m_{02}=-{\frac{emx^{*} \left( a\delta _{hb}+c \right) \left( bx^{*}+a \right) }{ \left( b \left( -\delta _{hb}\,y^{*}+1 \right) x^{*}+cy^{*}+a \right) ^{3}}},\\&m_{30}=-{\frac{em \left( \delta _{hb}\,y^{*}-1 \right) ^{3}{b}^{2} \left( cy^{*}+a \right) y^{*}}{ \left( b \left( -\delta _{hb}\,y^{*}+1 \right) x^{*}+cy^{*}+a \right) ^{4}}},\\&m_{21}=-{\frac{e \left( b\delta _{hb}\,x^{*}y^{*}-bx^{*}+2\,cy^{*}+2\,a \right) b \left( \delta _{hb} \,y^{*}-1 \right) \left( a\delta _{hb}+c \right) my^{*}}{ \left( -b\delta _{hb}\,x^{*}y^{*}+bx^{*}+cy^{*}+ a \right) ^{4}}}\\&\qquad -{\frac{em \left( \delta _{hb}\,y^{*}-1 \right) ^{2}b \left( cy^{*}+a \right) }{ \left( b \left( 1-\delta _{hb}\,y^{*} \right) x^{*}+cy^{*}+a \right) ^{3}}},\\&m_{12}=-{\frac{e \left( {x^{*}}^{2} \left( \delta _{hb}\,y^{*}-1 \right) {b}^{2}+2\,x^{*}y^{*} \left( a\delta _{hb}+c \right) b+a \left( cy^{*}+a \right) \right) \left( a \delta _{hb}+c \right) m}{ \left( b \left( -\delta _{hb}\,y^{*}+1 \right) x^{*}+cy^{*}+a \right) ^{4}}},\\&m_{03}=-{\frac{emx^{*} \left( b\delta _{hb}\,x^{*}-c \right) \left( a\delta _{hb}+c \right) \left( bx^{*}+a \right) }{ \left( b \left( -\delta _{hb}\,y^{*}+1 \right) x^{*}+cy^{*}+a \right) ^{4}}}. \end{aligned}$$