Bogdanov–Takens Bifurcation of Codimensions 3 and 4 in a Holling and Leslie type Predator–Prey System with Strong Allee Effect

Abstract

In this paper, we revisit a Leslie type predator–prey system with strong Allee effect on prey and simplified Holling type IV functional response proposed. Some more complex dynamical properties have been discovered. First, it is proved that the system exhibits a degenerate cusp of codimension 4 and a degenerate focus, saddle or elliptic of codimension 3 for different parameter values analytically. Further, various possible high codimensional bifurcation analyses are performed. It is shown that the system undergoes degenerate focus, saddle or elliptic type Bogdanov–Takens bifurcation of codimension 3 and degenerate cusp type Bogdanov–Takens bifurcation of codimension 4 as the parameters vary. Moreover, the existence and spatial location of the limit cycles are explored by calculating Hopf bifurcation and homoclinic bifurcation surfaces. Numerical simulations are carried out, and it is observed that three limit cycles coexist.

Appendices

Appendix A

The coefficient expressions of system (6) are listed as follows.

$p_{1}=\dfrac{1}{2c(-m-1+\sqrt{(m-1)^{2}-4c})}((-2\,m^{3}+3mc+2\,m^{2}+3c+2\,m-2)\sqrt{(m-1)^{2}-4c} +2\,m^{4}-7\,m^{2}c-4\,m^{3}-2c^{2}+2mc+4\,m^{2}-7c-4\,m+2)$,

$p_{2}=-\sqrt{(m-1)^{2}-4c}$,

$p_{3}=-\dfrac{1}{c^{2}(-m-1+\sqrt{(m-1)^{2}-4c})^{2}}((2\,m^{5}-4\,m^{4}+13c^{2}-12\,m^{3}c+13mc^{2}+4\,m^{2}c+2\,m^{3} -12c+2\,m^{2}+4mc-4\,m+2)\sqrt{(m-1)^{2}-4c}-2\,m^{6}+6\,m^{5}-6\,m^{4}-2+8\,m^{2}c+4\,m^{3}-20mc+16\,m^{4}c-33\,m^{2}c^{2} -20\,m^{3}c-10mc^{2}-33c^{2}+10c^{3}-6\,m^{2}+16c+6\,m)$,

$p_{4}=\dfrac{-1}{c(-m-1+\sqrt{(m-1)^{2}-4c})^{2}}((m+1)\sqrt{(m-1)^{2}-4c}-(m-1)^{2} +5c)((m+1)\sqrt{(m-1)^{2}-4c}-(m-1)^{2}+2c)$,

$p_{5}=\dfrac{1}{2c^{3}(-m-1+\sqrt{(m-1)^{2}-4c})^{3}}(((-26\,m^{3}c+37mc^{2}+10\,m^{2}c+4\,m^{5}-8\,m^{4}+37c^{3}+10mc+4+4\,m^{3}+4\,m^{2} -26c-8\,m)\sqrt{(m-1)^{2}-4c}-4-4\,m^{6}+12\,m^{5}+34\,m^{4}c-81\,m^{2}c^{2}-44\,m^{3}c-14mc^{2}+38c^{3}+20\,m^{2}c-44mc-81c^{2} +8\,m^{3}-12\,m^{4}-12\,m^{2}+34c+12\,m)((m+1)\sqrt{(m-1)^{2}-4c}-(m-1)^{2}+2c))$,

$p_{6}=\dfrac{2}{c^{2}(-m-1+\sqrt{(m-1)^{2}-4c})^{3}}((-m^{3}+4mc+m^{2}+4c+m-1)\sqrt{(m-1)^{2}-4c}+m^{4}-6\,m^{2}c-2\,m^{3}+7c^{2} +4mc+2\,m^{2}-6c-2\,m+1)((m+1)\sqrt{(m-1)^{2}-4c}-(m-1)^{2}+2c)$,

$p_{7}=\dfrac{-1}{2c^{4}(-m-1+\sqrt{(m-1)^{2}-4c})^{4}}((4+4\,m^{3}-28c+4\,m^{2}-8\,m+4\,m^{5}-8\,m^{4}+45c^{2}+45c^{2}m +12c m^{2}-28c m^{3}+12mc+4c+m-1)\sqrt{(m-1)^{2}-4c}-4+36c m^{4}-93c^{2}m^{2}-48c m^{3}-6c^{2}m+52c^{3}-4\,m^{6}+12\,m^{5}-12\,m^{4}-93c^{2}+8\,m^{3}-12\,m^{2}+24c m^{2}-48mc+36c+12\,m)((m+1)\sqrt{(m-1)^{2}-4c}-(m-1)^{2}+2c)$,

$p_{8}=-\dfrac{((m+1)\sqrt{(m-1)^{2}-4c}-(m-1)^{2}+2c)^{2}}{c^{3}(-m-1+\sqrt{(m-1)^{2}-4c})^{4}}((-2\,m^{3}+9mc+2\,m^{2}+9c+2\,m-2)\sqrt{(m-1)^{2}-4c}+2\,m^{4} -13\,m^{2}c-4\,m^{3}+19c^{2}+10mc+4\,m^{2}-13c-4\,m+2)$,

$q_{1}=q_{3}=\dfrac{-q_{2}}{2}=\dfrac{-2c}{(m+1-\sqrt{(m-1)^{2}-4c})}$,

$q_{4}=q_{6}=\dfrac{-q_{5}}{2}=\dfrac{4c}{(m+1-\sqrt{(m-1)^{2}-4c})^{2}}$,

$q_{7}=q_{9}=\dfrac{-q_{8}}{2}=\dfrac{-8c}{(m+1-\sqrt{(m-1)^{2}-4c})^{3}}$,

$q_{10}=q_{12}=\dfrac{-q_{11}}{2}=\dfrac{16c}{(m+1-\sqrt{(m-1)^{2}-4c})^{4}}$.

Appendix B

The coefficient expressions of system (9) are listed as follows.

$\phi _{1}=c(p_{1}+p_{2})$, $\phi _{2}=-\dfrac{p_{2}+q_{3}}{c}$, $\phi _{3}=c(p_{3}+p_{4})$, $\phi _{4}=3p_{3}+2p_{4}+\dfrac{(p_{2}+2q_{3})(p_{1}+p_{2})}{c}$,

$\phi _{5}=-\dfrac{2cp_{4}+cq_{6}+p_{2}^{2}+p_{2}q_{3}}{c^{2}}$, $\phi _{6}=cp_{5}+cp_{6}-\dfrac{(p_{1}+p_{2})^{2}q_{3}}{c}$,

$\phi _{7}=4p_{5}+3p_{6}+\dfrac{(p_{2}^{2}+2p_{2}q_{3}+2cq_{6})(p_{1}+p_{2})}{c^{2}}+\dfrac{(p_{2}+2p_{3}+2p_{4})q_{3} +(2p_{1}+3p_{2})p_{4}}{c}$,

$\phi _{8}=-\dfrac{3p_{6}+q_{9}}{c}-\dfrac{3p_{2}p_{4}+p_{2}q_{6}+p_{4}q_{3}}{c^{2}}-\dfrac{p_{2}^{2}(p_{2}+q_{3})}{c^{3}}$.

Appendix C

The coefficient expressions of system (16) are listed as follows.

$g_{30}=p_{3}+p_{4}$, $g_{21}=-\dfrac{1}{c^{2}}(cp_{4}+p_{2}^{2}+p_{2}q_{3})$, $g_{40}=-\dfrac{1}{2c}(p_{3}+p_{4})(p_{2}+q_{3})+p_{5}+p_{6}$, $g_{31}=-\dfrac{1}{c^{3}}(c^{2}p_{6}-cp_{4}q_{3}-p_{2}^{3}-3p_{2}^{2}q_{3}-2p_{2}q_{3}^{2})$, $h_{11}=-p_{2}$, $h_{30}=c(p_{3}+p_{4})$, $h_{21}=\dfrac{p_{2}(p_{2}+q_{3})}{2c}-p_{4}$, $h_{12}=-\dfrac{1}{c^{2}}(cq_{6}+p_{2}q_{3}+q_{3}^{2})$, $h_{40}=-\dfrac{3}{2}p_{2}p_{3}-\dfrac{3}{2}p_{2}p_{4}-\dfrac{1}{2}p_{3}q_{3}-\dfrac{1}{2}p_{4}q_{3}+cp_{5}+cp_{6}$, $h_{31}=-\dfrac{1}{2c^{2}}(2c^{2}p_{6}-2cp_{2}p_{4}-2cp_{3}q_{3}-4cp_{4}q_{3}+p_{2}^{3}+2p_{2}^{2}q_{3}+p_{2}q_{3}^{2})$, $h_{22}=-\dfrac{1}{2c^{3}}(2c^{2}q_{9}-cp_{2}q_{6}+2cp_{4}q_{3}-3cq_{3}q_{6}-p_{2}^{2}q_{3}-6p_{2}q_{3}^{2}-5q_{3}^{3})$.

Appendix D

The coefficient expressions of system (19) are listed as follows.

$k_{00}=x_{{2}} ( -x_{{2}}+1 ) ( -m+x_{{2}} ) -\dfrac{ x_{{2}}^{2}}{a_{{0}}+\lambda _{{1}}+ ( b_{{0}}+\lambda _{{2}} ) x_{{2}}^{2}}$,

$k_{10}=x_{{2}} ( -x_{{2}}+1 ) + ( -2\,x_{{2}}+1 ) ( -m+x_{{2}} ) -\dfrac{x_{{2}} ( -x_{{2}}^{2}b_{{0 }}-x_{{2}}^{2}\lambda _{{2}}+a_{{0}}+\lambda _{{1}} ) }{ ( x_{{2}}^{2}b_{{0}}+x_{{2}}^{2}\lambda _{{2}}+a_{{0}}+\lambda _{{1}} ) ^{2}}$,

$k_{01}=-\dfrac{x_{{2}}}{a_{{0}}+\lambda _{{1}}+ ( b_{{0}}+\lambda _{{2}} ) x_{{2}}^{2}}$, $k_{11}=-\dfrac{-x_{{2}}^{2}b_{{0}}-x_{{2}}^{2}\lambda _{{2}}+a_{{0}}+ \lambda _{{1}}}{(x_{{2}}^{2}b_{{0}}+x_{{2}}^{2}\lambda _{{2}}+a_{{0}}+ \lambda _{{1}})^{2}}$,

$k_{20}=-3\,x_{{2}}+1+m+\dfrac{ ( b_{{0}}+\lambda _{{2}} ) x_{{2}} ^{2} ( -x_{2}^{2}b_{{0}}-x_{{2}}^{2}\lambda _{{2}}+3\,a_{{0 }}+3\,\lambda _{{1}} ) }{ ( x_{{2}}^{2}b_{{0}}+x_{{2}}^{ 2}\lambda _{{2}}+a_{{0}}+\lambda _{{1}} ) ^{3}}$,

$k_{30}=-1+\dfrac{x_{{2}} ( b_{{0}}+\lambda _{{2}} )}{ ( x_{{2}}^{2}b_{{0}}+x_{{2}}^{2 }\lambda _{{2}}+a_{{0}}+\lambda _{{1}} ) ^{4}} ( b_{{0 }}^{2}x_{{2}}^{4}+2\,b_{{0}}\lambda _{{2}}x_{{2}}^{4}+\lambda _{{2 }}^{2}x_{{2}}^{4}-6\,a_{{0}}b_{{0}}x_{{2}}^{2}-6\,a_{{0}}\lambda _ {{2}}x_{{2}}^{2}-6\,b_{{0}}\lambda _{{1}}x_{{2}}^{2}-6\,\lambda _{{1 }}\lambda _{{2}}x_{{2}}^{2}+{a_{{0}}}^{2}+2\,a_{{0}}\lambda _{{1}}+ \lambda _{{1}}^{2} )$,

$k_{21}={\dfrac{x_{{2}} ( b_{{0}}+\lambda _{{2}} ) ( -x_{{2} }^{2}b_{{0}}-x_{{2}}^{2}\lambda _{{2}}+3\,a_{{0}}+3\,\lambda _{{1}} ) }{ ( x_{{2}}^{2}b_{{0}}+x_{{2}}^{2}\lambda _{{2}}+a_{ {0}}+\lambda _{{1}} ) ^{3}}}$, $l_{10}=c_{0}+\lambda _{3}$, $l_{01}=-c_{0}-\lambda _{3}$, $l_{20}=-\dfrac{c_{0}+\lambda _{3}}{x_{2}}$, $l_{11}=\dfrac{2(c_{0}+\lambda _{3})}{x_{2}}$, $l_{02}=-\dfrac{c_{0}+\lambda _{3}}{x_{2}}$, $l_{30}=\dfrac{c_{0}+\lambda _{3}}{x_{2}^{2}}$, $l_{21}=-\dfrac{2(c_{0}+\lambda _{3})}{x_{2}^{2}}$, $l_{12}=\dfrac{c_{0}+\lambda _{3}}{x_{2}^{2}}$.

Appendix E

The coefficient expressions of system (20) are listed as follows.

$m_{00}=\dfrac{k_{{00}} ( k_{{00}}l_{{02}}-k_{{01}}l_{{01}} ) }{k_{{01}}}$,

$m_{10}=\dfrac{l_{{12}}k_{{00}}^{2}k_{{01}}-k_{{00}}^{2}k_{{11}}l_{{02}}-l_{{ 11}}k_{{00}}k_{{01}}^{2}+2\,k_{{00}}k_{{01}}k_{{10}}l_{{02}}+l_{{10}}k_{ {01}}^{3}-k_{{01}}^{2}k_{{10}}l_{{01}}}{k_{{01}}^{2}}$,

$m_{01}=-\dfrac{k_{{00}}k_{{11}}+2\,k_{{00}}l_{{02}}-k_{{01}}k_{{10}}-k_{{01}}l_{{ 01}}}{k_{{01}}}$,

$m_{20}=-\dfrac{1}{k_{01}^{3}}(k_{{00}}^{2}k_{{01}}k_{{11}}l_{{12}}+k_{{00}}^{2}k_{{01}}k_{{ 21}}l_{{02}}-k_{{00}}^{2}k_{{11}}^{2}l_{{02}}+l_{{21}}k_{{00}}k_{{01}} ^{3}-2\,k_{{00}}k_{{01}}^{2}k_{{10}}l_{{12}}-2\,k_{{00}}k_{{01}}^{2}k _{{20}}l_{{02}}+2\,k_{{00}}k_{{01}}k_{{10}}k_{{11}}l_{{02}}-l_{{20}}k_{{01 }}^{4}+k_{{01}}^{3}k_{{10}}l_{{11}}-k_{{01}}^{3}k_{{11}}l_{{10}}+k _{{01}}^{3}k_{{20}}l_{{01}}-k_{{01}}^{2}k_{{10}}^{2}l_{{02}})$,

$m_{11}=-\dfrac{1}{k_{{01}}^{2}}(2\,k_{{00}}k_{{21}}k_{{01}}+2\,l_{{12}}k_{{00}}k_{{01}}-k_{{00}}k_{{11}}^{2}-2\,k_{{00}} k_{{11}}l_{{02}}-2\,k_{{20}}k_{{01}}^{2}-l_{{11 }}k_{{01}}^{2}+k_{{01}}k_{{10}}k_{{11}}+2\,k_{{01}}k_{{10}}l_{{02}})$, $m_{02}=\dfrac{l_{{02}}+k_{{11}}}{k_{{01}}}$,

$m_{30}=-\dfrac{1}{k_{{01}}^{4}}(k_{{00}}^{2}k_{{01}}^{2}k_{{21}}l_{{12}}-k_{{00}}^{2}k_{{01 }}k_{{11}}^{2}l_{{12}}-2\,k_{{00}}^{2}k_{{01}}k_{{11}}k_{{21}}l_{{02} }+k_{{00}}^{2}k_{{11}}^{3}l_{{02}}-2\,k_{00}k_{{01}}^{3}k_{{20}}l_ {{12}}-2\,k_{{00}}k_{{01}}^{3}k_{{30}}l_{{02}}+2\,k_{{00}}k_{{01}}^{2}k _{{10}}k_{{11}}l_{{12}}+2\,k_{{00}}k_{{01}}^{2}k_{{10}}k_{{21}}l_{{02}} +2\,k_{{00}}k_{{01}}^{2}k_{{11}}k_{{20}}l_{{02}}-2\,k_{{00}}k_{{01}}k_{{ 10}}k_{{11}}^{2}l_{{02}}-l_{{30}}k_{{01}}^{5}+k_{{01}}^{4}k_{{10}}l _{{21}}-k_{{01}}^{4}k_{{11}}l_{{20}}+k_{{01}}^{4}k_{{20}}l_{{11}}-k _{{01}}^{4}k_{{21}}l_{{10}}+k_{{01}}^{4}k_{{30}}l_{{01}}-k_{{01}}^{3} k_{{10}}^{2}l_{{12}}-2\,k_{{01}}^{3}k_{{10}}k_{{20}}l_{{02}}+k_{{01} }^{2}k_{{10}}^{2}k_{{11}}l_{{02}})$,

$m_{21}=\dfrac{1}{k_{{01}}^{3}}(3\,k_{{00}}k_{{01}}k_{{11}}k_{{21}}+2\,k_{{00}}k_{{01}}k_{{11}}l_{ {12}}+2\,k_{{00}}k_{{01}}k_{{21}}l_{{02}}-k_{{00}}k_{{11}}^{3}-2\,k_{{00} }k_{{11}}^{2}l_{{02}}+3\,k_{{01}}^{3}k_{{30}}+l_{{21}}k_{{01}}^{3}- 2\,k_{{01}}^{2}k_{{10}}k_{{21}}-2\,k_{{01}}^{2}k_{{10}}l_{{12}}-k_{ {01}}^{2}k_{{11}}k_{{20}}-2\,k_{{01}}^{2}k_{{20}}l_{{02}}+k_{{01}}k_{{ 10}}k_{{11}}^{2}+2\,k_{{01}}k_{{10}}k_{{11}}l_{{02}})$,

$m_{12}=\dfrac{2\,k_{{21}}k_{{01}}+l_{{12}}k_{{01}}-k_{{11}}^{2}-l_{{02}}k_{{ 11}}}{k_{{01}}^{2}}$.