Complex dynamics of a prey-predator interaction model with Holling type-II functional response incorporating the effect of fear on prey and non-linear predator harvesting

Abstract

In the present article, we have investigated the impact of the fear effect and non-linear predator harvesting in a prey-predator interaction model with Holling type-II functional response. Fear for predator population enhances the survival probability of prey population, and it can greatly reduce the reproduction of prey population. We have investigated all the biologically feasible equilibrium points and the positivity and boundedness of the system solutions. We have analyzed the local and global stability of the feasible equilibrium points in terms of the model parameters. Analytically we have established that the intrinsic growth rate of the prey population can stabilize the prey-predator interactions by ignoring the existence of periodic behaviors. The model system undergoes through Transcritical, Saddle-Node, Hopf bifurcation by considering the intrinsic growth rate of the prey population as a bifurcation parameter and Bogdanov–Takens bifurcation with respect to the prey intrinsic growth rate and rate of predator harvesting. Numerically we have identified two parametric planes, which are divided into sub-regions associated with different numbers and nature of the equilibrium points by various bifurcation curves. We also found that the system may exhibit bi-stability behavior by producing stable axial and interior equilibrium points. Lastly, the manuscript is concluded with some recommendations.

Appendices

Appendices

1.1 Appendix 1

$A_1=Eac^2kp^2qr_1r_2-2Eackmpq^2r_1r_2+Eakm^2q^3r_1r_2-Eacknpq^2r_2+Eakmnq^3r_2-ac^2p^2qr_2^2+2acmpq^2r_2^2-am^2q^3r_2^2 ;\, A_2=Ec^2dkp^2qr_1r_2-2Ecdkmpq^2r_1r2+Edkm^2q^3r_1r_2+Eac^2kp^2r_1r_2-4Eackmpqr_1r_2+3Eakm^2q^2r_1r_2-Ecdknpq^2r_2+Edkmnq^3r_2-2Eacknpqr_2+3Eakmnq^2r_2-c^2dp^2qr_2^2+c^2p^2qrr_2^2+2cdmpq^2r_2^2-2cmpq^2rr_2^2-dm^2q^3r_2^2+m^2q^3rr_2^2-ac^2p^2r_2^2+4acmpqr_2^2-3am^2q^2r_2^2 ;\, A_3=-E^2c^2kp^3r_1^2+2E^2ckmp^2qr_1^2-E^2km^2pq^2r_1^2+2E^2cknp^2qr_1-2E^2kmnpq^2r_1+Ec^2dkp^2r_1r_2-4Ecdkmpqr_1r_2+3Edkm^2q^2r_1r_2-E^2kn^2pq^2-2Eackmpr_1r_2+3Eakm^2qr_1r_2+Ec^2p^3r_1r_2-2Ecdknpqr_2-2Ecmp^2qr_1r_2+3Edkmnq^2r_2+Em^2pq^2r_1r_2-Eacknpr_2+3Eakmnqr_2-Ecnp^2qr_2+Emnpq^2r_2-c^2dp^2r_2^2+c^2p^2rr_2^2+4cdmpqr_2^2-4cmpqrr_2^2-3dm^2q^2r_2^2+3m^2q^2rr_2^2+2acmpr_2^2-3am^2qr_2^2 ;\, A_4=2E^2ckmp^2r_1^2-2E^2km^2pqr_1^2+2E^2cknp^2r_1-4E^2kmnpqr_1-2Ecdkmpr_1r_2+3Edkm^2qr_1r_2-2E^2kn^2pq+Eakm^2r_1r_2-Ecdknpr_2-2Ecmp^2r_1r_2+3Edkmnqr_2+2Em^2pqr_1r_2+Eakmnr_2-Ecnp^2r_2+2Emnpqr_2+2cdmpr_2^2-2cmprr_2^2-3dm^2qr_2^2+3m^2qrr_2^2-am^2r_2^2 ;\, A_5=-E^2km^2pr_1^2-2E^2kmnpr_1+Edkm^2r_1r_2-E^2kn^2p+Edkmnr_2+Em^2pr_1r_2+Emnpr_2-dm^2r_2^2+m^2rr_2^2$.

1.2 Appendix 2

$a_{11}=-\dfrac{1}{(ky_1+1)(qx_1+1)^2}(2akq^2x_1^3y_1+dkq^2x_1^2y_1+4akqx_1^2y_1+2aq^2x_1^3+2dkqx_1y_1+dq^2x_1^2-q^2rx_1^2+2akx_1y_1+4aqx_1^2+kpy_1^2+dky_1+2dqx_1-2q_rx_1+2ax_1+py_1+d-r)$ $a_{12}=-\dfrac{rx_1k}{(ky_1+1)^2}-\dfrac{px_1}{(qx_1+1)}$ $a_{13}=-a+\dfrac{py_1q}{(qx_1+1)^3}$ $a_{14}=-\dfrac{(kq^2rx_1^2+k^2py_1^2+2kqrx_1+2kpy_1+kr+p)}{(qx_1+1)^2(ky_1+1)^2)}$ $a_{15}=\dfrac{rx_1k^2}{(ky_1+1)^3}$, $a_{16}=-\dfrac{pq^2y_1}{(qx_1+1)^4}$, $a_{17}=\dfrac{pq(qx_1+1)}{(qx_1+1)^4}$, $a_{18}=\dfrac{rk^2}{(ky_1+1)^3}$, $a_{19}=-\dfrac{rx_1k^3}{(ky_1+1)^4}$, $a_{21}=-\dfrac{cpy_1}{(qx_1+1)^2}$, $a_{22}=\dfrac{1}{(qx_1+1)(Er_1+r_2y_1)^2}(E^2cpr_1^2x_1-E^2mqr_1^2x_1+2Ecpr_1r_2x_1y_1-2Emqr_1r_2x_1y_1+cpr_2^2x_1y_1^2-mqr_2^2x_1y_1^2-E^2nqr_1x_1-E^2mr_1^2-2Emr_1r2y_1-mr_2^2y_1^2-E^2nr_1)$ $a_{23}=-\dfrac{cpy_1q}{(qx_1+1)^3}$ $a_{24}=\dfrac{cp}{(qx_1+1)^2}$, $a_{25}=\dfrac{nE^2r_1r_2}{(Er_1+r2y_1)^3}$, $a_{26}=\dfrac{cpq^2y_1}{(qx_1+1)^4}$, $a_{27}=-\dfrac{cpq(qx_1+1)}{(qx_1+1)^4}$, $a_{28}=0$, $a_{29}=-\dfrac{nE^2r_1r_2^2}{(Er_1+r_2y_1)^4}$, $B_1=(a_{13}a_{12}^2-a_{14}a_{12}a_{11})u^2+\omega _2(2a_{15}a_{11}-a_{14}a_{12})uv+\omega _2^2a{15}v^2+(a_{18}a_{12}a_{11}^2-a_{19}a_{11}^3+a_{16}a_{12}^3-a_{17}a_{12}^2a_{11})u^3 +\omega _2(2a_{18}a_{11}a_{12}-a_{17}a_{12}^2-3a_{19}a_{11}^2)u^2v+\omega _2^2(a_{18}a_{12}-3a_{19}a_{11})uv^2-\omega _2^3a_{19}v^3$ $B_2=(a_{23}a_{22}^2-a_{24}a_{22}a_{21})u^2+\omega _2(2a_{25}a_{21}-a_{24}a_{22})uv+\omega _2^2a{25}v^2+(a_{28}a_{22}a_{21}^2-a_{29}a_{21}^3+a_{26}a_{22}^3-a_{27}a_{22}^2a_{21})u^3 +\omega _2(2a_{28}a_{21}a_{22}-a_{27}a_{22}^2-3a_{29}a_{21}^2)u^2v+\omega _2^2(a_{28}a_{22}-3a_{29}a_{21})uv^2-\omega _2^3a_{29}v^3$

1.3 Appendix 3

$d_{11}(\psi )=\dfrac{r^{BT}+\psi _1}{ky^*+1)}-d-2ax^*-\dfrac{py^*}{(qx^*+1)^2}, \ d_{12}(\psi )=-\dfrac{(r^{BT}+\psi _1)kx^*}{(ky^*+1)^2}-\dfrac{px^*}{(qx^*+1)}, l_{00}(\psi )=\dfrac{(r^{BT}+\psi _1)x^*}{(ky^*+1)}-dx^*-ax^{*2}-\dfrac{px^*y^*}{(qx^*+1)}, \ l_{20}(\psi )=-a+\dfrac{pqy^*}{(qx^*+1)^3}, l_{11}(\psi )=-\dfrac{(r^{BT}+\psi _1)k}{(ky^*+1)^2}-\dfrac{p}{(qx^*+1)^2}, \ l_{02}(\psi )=\dfrac{(r^{BT}+\psi _1)k^2x^*}{(ky^*+1)^3}, \ d_{21}(\psi )=\dfrac{cpy^*}{(qx^*+1)^2}, m_{00}(\psi )=\dfrac{cpx^*y^*}{qx^*+1}-my^*-\dfrac{n(E^{BT}+\psi _2)y^*}{(r_1(E^{BT}+\psi _2)+r_2y^*)}, \ m_{20}(\psi )=-\dfrac{cpqy^*}{(qx^*+1)^3}, m_{11}(\psi )=\dfrac{cp}{(qx^*+1)^2}, d_{22}(\psi )={nyr_2E^{BT}}{(r_1E^{BT}+r_1\psi _2+r_2y^*)^2}+\dfrac{nr_2\psi _2y*}{(r_1E^{BT}+r_1\psi _2+r_2y^*)^2}\dfrac{cpx^*}{(qx^*+1)}-m-{nE^{BT}}{(r_1E^{BT}+ r_1\psi _2+r_2y^*)} -{n\psi _2}{(r_1E^{BT}+r_1\psi _2+r_2y^*)}, \ m_{02}(\psi )=\dfrac{n(E^{BT}+\psi _2)^2r_1r_2}{((r_1E^{BT}+r_1\psi _2+r_2y^*)^2(r_1(E^{BT}+\psi _2)+r_2y^*))}$

1.4 Appendix 4

$e_{02}(\psi )=l_{20}-\dfrac{l_{11}d_{11}}{d_{12}}+\dfrac{l_{02}d_{11}^2}{d_{12}^2}$, $e_{11}(\psi )=\dfrac{l_{11}}{d_{12}}-\dfrac{2d_{11}l_{02}}{d_{12}^2}$, $e_{02}(\psi )=\dfrac{l_{02}}{d_{11}^2}$, $n_{00}(\psi )=d_{11}l_{00}+d_{12}m_{00}$, $n_{10}(\psi )=d_{12}d_{21}-d_{11}d_{22}$, $n_{01}(\psi )=d_{11}+d_{22}$, $n_{20}(\psi )=d_{11}l_{20}-\dfrac{l_{11}d_{11}^2}{d_{12}}+\dfrac{l_{02}d_{11}^3}{d_{12}^2}+d_{12}m_{20}-d_{11}m_{11}+\frac{m_{02}d_{11}^2}{d_{12}}$, $n_{11}(\psi )=m_{11}+\dfrac{l_{11}d_{11}}{d_{12}}-\dfrac{2d_{11}^2l_{02}}{d_{12}^2}-\dfrac{2d_{11}m_{02}}{d_{12}}$, $n_{02}(\psi )=\dfrac{d_{11}l_{02}}{d_{12}^2}+\dfrac{m_{02}}{d_{12}}$

1.5 Appendix 5

$f_{00}(\psi )=n_{00}(\psi )-l_{00}(\psi )n_{01}(\psi )+l_{00}^2(\psi )(n_{02}(\psi )-d_{02}(\psi )n_{01}(\psi ))-2l_{00}(\psi )d_{02}(\psi )(n_{00}(\psi )-l_{00}(\psi )n_{01}(\psi ))+...=n_{00}(\psi )-l_{00}(\psi )n_{01}(\psi )+l_{00}^2d_{02}(\psi )n_{01}(\psi )+l_{00}^2(\psi )n_{02}(\psi )+... f_{10}(\psi )=n_{10}(\psi )+l_{00}(\psi )d_{11}(\psi )n_{01}(\psi )-l_{00}(\psi )n_{11}(\psi )+d_{11}(\psi )(n_{00}(\psi )-l_{00}(\psi )n_{01}(\psi )) -2l_{00}(\psi )d_{02}(\psi )n_{10}(\psi )+...=n_{10}(\psi )+d_{11}(\psi )n_{00}(\psi )-l_{00}(\psi )n_{11}(\psi )-2l_{00}(\psi )d_{02}(\psi )n_{10}(\psi )+..., f_{01}(\psi )=n_{01}(\psi )-l_{00}(\psi )d_{11}(\psi )+2l_{00}(\psi )d_{02}(\psi )n_{01}(\psi )-2l_{00}(\psi )n_{02}(\psi )+2d_{02}(\psi )n_{00}(\psi )+..., g_{20}(\psi )=n_{20}(\psi )+d_{11}(\psi )n_{10}(\psi )-n_{01}(\psi )d_{20}(\psi )+..., f_{11}(\psi )=-d_{11}(\psi )n_{01}(\psi )+n_{11}(\psi )+2d_{20}(\psi )+d_{11}(\psi )n_{01}(\psi )+2d_{02}(\psi )n_{10}(\psi )+..., =n_{11}(\psi )+2d_{20}(\psi )+2d_{02}(\psi )n_{10}(\psi )+..., f_{02}(\psi )=n_{02}(\psi )-d_{02}(\psi )n_{01}(\psi )+d_{11}(\psi )+2d_{02}(\psi )n_{01}(\psi )+....$