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Study of co-dimension two bifurcation of a prey–predator model with prey refuge and non-linear harvesting on both species

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Abstract

The dynamics of prey–predator system, when one or both the species are harvested non-linearly, has become a topic of intense study because of its wide applications in biological control and species conservation. In this paper we have discuss different bifurcation analysis of a two dimensional prey–predator model with Beddington–DeAngelis type functional response in the presence of prey refuge and non-linear harvesting of both species. We have studied the positivity and boundedness of the model system. All the biologically feasible equilibrium points are investigated and their local stability is analyzed in terms of model parameters. The global stability of coexistence equilibrium point has been discussed. Depending on the prey harvesting effort (\(E_1\)) and degree of competition among the boats, fishermen and other technology (\(l_1\)) used for prey harvesting, the number of axial and interior equilibrium points may change. The system experiences different type of co-dimension one bifurcations such as transcritical, Hopf, saddle-node bifurcation and co-dimension two Bogdanov–Takens bifurcation. The parameter values at the Bogdanov–Takens bifurcation point are highly sensitive in the sense that the nature of coexistence equilibrium point changes dramatically in the neighbourhood of this point. The feasible region of the bifurcation diagram in the \(l_1-E_1\) parametric plane divides into nine distinct sub-regions depending on the number and nature of equilibrium points. We carried out some numerical simulations using the Maple and MATLAB software to justify our theoretical findings and finally some conclusions are given.

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Acknowledgements

The authors would like to thank the anonymous reviewers for their careful reading, useful comments and constructive suggestions for the improvement of the manuscript of the present research work.

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Authors and Affiliations

  1. Department of Applied Mathematics, University of Calcutta, Kolkata, 700009, India

    Prahlad Majumdar, Uttam Ghosh, Susmita Sarkar & Surajit Debnath

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  1. Prahlad Majumdar
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Correspondence to Surajit Debnath.

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Appendices

Appendices

Appendix I

The coefficient of six degree polynomial equation (6) is given by MAPLE software in the following:

$$\begin{aligned} a_0{} & {} =abcmp_1^2p_2-abcp_1^2p_2,\\ a_1{} & {} =2E_1abc l_1mp_1p_2+a^2dm^2p_1^2p_2-2E_1abc l_1p_1p_2\\{} & {} +E_2b^2c l_2p_1^2-2a^2dmp_1^2p_2-2abcmp_1^2p_2\\{} & {} -acm^2p_1^2p_2+a^2dp_1^2p_2+2abcp_1^2p_2\\{} & {} +2acmp_1^2p_2-acp_1^2 p_2-bcp_1^2p_2,\\ a_2{} & {} =E_1^2abc l_1^2mp_2+2E_1a^2dl_1m^2p_1p_2-E_1^2abc l_1^2p_2\\{} & {} +2E_1E_2b^2c l-1 l_2p_1-4E_1a^2dl_1mp_1p_2\\{} & {} -4E_1abc l_1mp_1p_2-2E_1ac l_1m^2p_1p_2+E_2abd l_2mp_1^2\\{} & {} -a^2dm^2p_1^2p_2+2E_1a^2d l_1 p_1p_2\\{} & {} +4E_1abc l_1p_1p_2+2E_1abcmp_1p_2+4E_1ac l_1mp_1p_2-E_2abd l_2p_1^2\\{} & {} -2E_2b^2c l_2p_1^2-2E_2bc l_2mp_1^2\\{} & {} +2a^2dmp_1^2p_2+abcmp_1^2p_2+acm^2p_1^2p_2-2E_1abcp_1p_2\\{} & {} -2E_1ac l_1p_1p_2-2E_1bc l_1*p_1p_2+E_2abmp_1^2\\{} & {} +2E_2bc l_2p_1^2-a^2dp_1^2p_2-abcp_1^2p_2-2acmp_1^2p_2\\{} & {} -2admp_1^2p_2-E_2abp_1^2+acp_1^2p_2\\{} & {} +2adp_1^2p_2+2bcp_1^2p_2+cmp_1^2p_2-cp_1^2p_2,\\ a_3{} & {} =E_1^2a^2d l_1^2m^2p_2+E_1^2E_2b^2c l_1^2 l_2-2E_1^2a^2d l_1^2mp_2\\{} & {} -2E_1^2abcl_1^2mp_2-E_1^2ac l_1^2m^2p_2\\{} & {} +2E_1E_2abd l_1 l_2mp_1-2E_1a^2d l_1m^2p_1p_2+E_1^2a^2d l_1^2p_2\\{} & {} +2E_1^2abc l_1^2p_2+2E_1^2abc l_1mp_2\\{} & {} +2E_1^2ac l_1^2mp_2-2E_1E_2abd l_1 l_2p_1-4E_1E_2b^2c l_1 l_2p_1\\{} & {} -4E_1E_2bc l_1 l_2mp_1+4E_1a^2d l_1mp_1p_2\\{} & {} +E_1a^2dm^2p_1p_2+2E_1abc l_1mp_1p_2+2E_1ac l_1m^2p_1p_2-E_2abd l_2mp_1^2\\{} & {} -E_2ad l_2m^2p_1^2-2E_1^2abc l_1 p_2\\{} & {} -E_1^2ac l_1^2p_2-E_1^2bc l_1^2p_2+2E_1E_2abl_1mp_1+2E_1E_2b^2c l_2p_1\\{} & {} +4E_1E_2bc l_1 l_2 p_1-2E_1a^2d l_1p_1p_2\\{} & {} -2E_1a^2dmp_1p_2-2E_1abc l_1p_1p_2-2E_1abcmp_1p_2-4E_1acl_1mp_1p_2\\{} & {} -E_1acm^2p_1p_2-4E_1ad l_1mp_1p_2\\{} & {} +E_2abd l_2p_1^2+2E_2adl_2mp_1^2+E_2b^2cl_2p_1^2+2E_2bcl_2mp_1^2\\{} & {} +E_2c l_2m^2p_1^2-2E_1E_2ab l_1p_1+E_1a^2dp_1p_2\\{} & {} +2E_1abcp_1p_2+2E_1acl_1p_1p_2+2E_1acmp_1p_2+4E_1adl_1p_1p_2\\{} & {} +4E_1bcl_1p_1p_2+2E_1c l_1mp_1p_2-E_2abmp_1^2-E_2adl_2p_1^2\\{} & {} -E_2am^2p_1^2-2E_2bc l_2p_1^2-E_2bdl_2p_1^2-2E_2c l_2p_1^2+2admp_1^2p_2\\{} & {} -E_1acp_1p_2-2E_1bcp_1p_2-2E_1c l_1p_1p_2+E_2abp_1^2+2E_2amp_1^2\\{} & {} +E_2c l_2p_1^2-2adp_1^2p_2-bcp_1^2p_2-cmp_1^2p_2\\{} & {} -E_2ap_1^2-E_2bp_1^2+cp_1^2p_2+dp_1^2p_2, \\ a_4{} & {} =E_1^2E_2abd l_1^2 l_2 m-E_1^2a^2d l_1^2 m^2p_2\\{} & {} -E_1^2E_2abd l_1^2 l_2-2E_1^2E_2b^2c l_1^2 l_2\\{} & {} -2E_1^2E_2bc l_1^2 l_2 m+2E_1^2a^2d l_1^2 mp_2+E_1^2a^2d l_1 m^2p_2\\{} & {} +E_1^2abc l_1^2 mp_2+E_1^2ac l_1^2 m^2p_2\\{} & {} -2E_1E_2abd l_1 l_2 mp_1-2E_1E_2ad l1 l_2 m^2p_1+E_1^2E_2abl_1^2m\\{} & {} +2E_1^2E_2b^2c l_1 l_2+2E_1^2E_2bc l_1^2 l_2\\{} & {} -E_1^2a^2d l_1^2p_2-2E_1^2a^2d l_1 mp_2-E_1^2abc l_1^2p_2-2E_1^2abc l_1 mp_2\\{} & {} -2E_1^2ac l_1^2 mp_2-E_1^2ac l_1 m^2p_2\\{} & {} -2E_1^2ad l_1^2 mp_2+2E_1E_2abd l_1 l_2 p_1+E_1E_2abd l_2 mp_1\\{} & {} +4E_1E_2ad l_1 l_2 mp_1+2E_1E_2b^2c l_1 l_2 p_1+4E_1E_2bc l_1 l_2 mp_1\\{} & {} +2E_1E_2c l_1 l_2 m^2p_1-E_1^2E_2ab l_1^2+E_1^2a^2dl_1p_2+2E_1^2abc l_1 p_2\\{} & {} +E_1^2abcmp_2+E_1^2ac l_1^2 p_2+2E_1^2ac l_1mp_2\\{} & {} +2E_1^2ad l_1^2 p_2+2E_1^2 bc l_1^2p_2+E_1^2c l_1^2 mp_2\\{} & {} -E_1E_2abd l_2 p_1-2E_1E_2abl_1mp_1-2E_1E_2ad l_1 l_2p_1\\{} & {} -2E_1E_2a l_1m^2p_1-2E_1E_2b^2c l_2 p_1-4E_1E_2bc l_1 l_2 p_1\\{} & {} -2E_1E_2bc l_2 mp_1-2E_1E_2bd l_1 l_2p_1-4E_1E_2c l_1 l_2 mp_1\\{} & {} +4E_1ad l_1 mp_1p_2-E_1^2abcp_2-E_1^2ac l_1 p_2-2E_1^2bc l_1p_2\\{} & {} -E_1^2c l_1^2 p_2+2E_1E_2abl_1p_1+E_1E_2abmp_1+4E_1E_2a l_1 mp_1\\{} & {} +2E_1E_2bc l_2 p_1+2E_1E_2c l_1 l_2 p_1-4E_1adl_1p_1p_2-2E_1admp_1p_2\\{} & {} -2E_1bc l_1p_1p_2-2E_1cl_1mp_1p_2+E_2bd l_2 p_1^2\\{} & {} +E_2d l_2 mp_1^2-E_1E_2abp_1-2E_1E_2a l_1 p_1-2E_1E_2bl_1p_1+2E_1adp_1p_2\\{} & {} +2E_1bcp_1p_2+2E_1c l_1 p_1p_2+E_1cmp_1p2\\{} & {} +2E_1d l_1p_1p_2-E_2d l_2p_1^2-E_1cp_1p_2+E_2bp_1^2+E_2mp_1^2-dp_1^2p_2-E_2p_1^2, \\ a_5{} & {} =-E_1^2E_2abdl_1^2l_2m-E_1^2E_2adl_1^2l_2m^2+E_1^2E_2abdl_1^2l_2\\{} & {} +E_1^2E_2abd l_1 l_2 m+2E_1^2E_2ad l_1^2 l_2 m\\{} & {} +E_1^2 E_2b^2c l_1^2 l_2+2E_1^2 E_2bc l_1^2 l_2 m+E_1^2E_2c l_1^2 l_2m^2\\{} & {} -E_1^2E_2abd l_1 l_2-E_1^2E_2ab l_1^2 m\\{} & {} -E_1^2E_2 ad l_1^2 l_2-E_1^2E_2a l_1^2m^2-2E_1^2E_2b^2c l_1 l_2\\{} & {} -2E_1^2*E_2bc l_1^2 l_2-2E_1^2 E2bc l_1 l_2m-E_1^2E_2bd l_1^2 l_2\\{} & {} -2E_1^2E_2c l_1^2 l_2 m+2E_1^2ad l_1^2 mp_2+E_1^2E_2abl_1^2+E_1^2E_2ab l_1 m\\{} & {} +2E_1^2E_2a l_1^2 m+E_1^2E_2b^2c l_2+2E_1^2E_2bcl_1l_2\\{} & {} +E_1^2E_2c l_1^2l_2-2E_1^2adl_1^2p_2-2E_1^2adl_1mp_2-E_1^2bcl_1^2p_2\\{} & {} -E_1^2c l_1^2 mp_2+2E_1E_2bdl_1l_2p_1+2E_1E_2dl_1 l_2mp_1\\{} & {} -E_1^2E_2abl_1-E_1^2E_2a l_1^2-E_1^2E_2bl_1^2+2E_1^2ad l_1 p_2\\{} & {} +2E_1^2bcl_1p_2+E_1^2cl_1^2p_2+E_1^2c l_1mp_2+E_1^2dl_1^2p_2\\{} & {} -E_1E_2bd l_2 p_1-2E_1E_2 d l_1 l_2p_1-E_1^2bcp_2\\{} & {} -E_1^2cl_1p_2+2E_1 E_2 bl_1p_1+2E_1E_2 l_1 mp_1\\{} & {} -2E_1d l_1 p_1p_2-E_1E_2bp_1-2E_1E_2 l_1 p_1+E_1dp_1p_2, \\ a_6{} & {} =E_1^2E_2bd l_1^2 l_2+E_1^2E_2d l_1^2l_2 m-E_1^2E_2bd l_1 l_2\\{} & {} -E_1^2E_2 d l_1^2 l_2+E_1^2E_2b l_1^2+E_1^2E_2 l_1^2 m\\{} & {} -E_1^2d l_1^2p_2-E_1^2E_2bl_1-E_1^2E_2 l_1^2+E_1^2dl_1p_2.\\ \end{aligned}$$

Appendix II

$$\begin{aligned} h_1{} & {} =-2+\frac{2(1-m)^2y^*a}{(1+a(1-m)x^*+by^*)^2}-\frac{2(1-m)^3x^*y^*a^2}{(1+a(1-m)x^*+by^*)^3}\\{} & {} +\frac{2E_1p_1}{(E_1 l_1+p_1x^*)^2}-\frac{2E_1x^*p_1^2}{(E_1 l_1+p_1x^*)^3},\\ h_2{} & {} =-\frac{(1-m)}{(1+a(1-m)x^*+by^*)}+\frac{(1-m)y^*b}{(1+a(1-m)x^*+by^*)^2}\\{} & {} +\frac{(1-m)^2 x^*a}{(1+a(1-m)x^*+by^*)^2}-\frac{2(1-m)^2 x^*y^*ab}{(1+a(1-m)x^*+by^*)^3},\\ h_3{} & {} =\frac{(2(1-m))x^*b}{(1+a(1-m)x^*+by^*)^2}-\frac{(2(1-m))x^*y^*b^2}{(1+a(1-m)x^*+by^*)^3},\\ h_4{} & {} =-\frac{2c(1-m)^2y^*a}{(1+a(1-m)x^*+by^*)^2}+\frac{2c(1-m)^3 x^*y^*a^2}{(1+a(1-m)x^*+by^*)^3}, \\ h_5{} & {} =\frac{c(1-m)}{(1+a(1-m)x^*+by^*)}-\frac{c(1-m)y^*b}{(1+a(1-m)x^*+by^*)^2}\\{} & {} -\frac{c(1-m)^2 x^*a}{(1+a(1-m)x^*+by^*)^2}+\frac{2c(1-m)^2 x^*y^*ab}{(1+a(1-m)x^*+by^*)^3},\\ h_6{} & {} =-\frac{2c(1-m)x^*b}{(1+a(1-m)x^*+by^*)^2}+\frac{2c(1-m)x^*y^*b^2}{(1+a(1-m)x^*+by^*)^3}\\{} & {} +\frac{2E_2p_2}{(E_2l_2+p_2y^*)^2}-\frac{2E_2y^*p_2^2}{(E_2l_2+p_2y^*)^3}.\\ \end{aligned}$$

Appendix III

$$\begin{aligned} \gamma _{10}{} & {} =1-2x^*-\frac{(1-m)y^*(1+by^*)}{(1+a(1-m)x^*+by^*)^2}-\frac{E_1^2 l_1}{(l_1 E_1+p_1 x^*)^2}, \\ \gamma _{01}{} & {} =-\frac{(1-m)x^*[1+(1-m)ax^*]}{(1+a(1-m)x^*+by^*)^2},\\ \gamma _{20}{} & {} =-1+\frac{(-1+m)(-by^{*2}-y^*)a(1-m)}{(amx^*-ax^*-by^*-1)^2(1+a(1-m)x^*+by^*)}\\{} & {} +\frac{E_1^2 l_1 p_1}{(E_1 l_1+x^*p_1)^3},\\ \gamma _{11}{} & {} =-\frac{(1-m-\frac{(-1+m)(amx^{*2}-ax^{*2}-x^*)a(1-m)}{(amx^*-ax^*-by^*-1)^2}-\frac{(-1+m)(-by^{*2}-y^*)b}{(amx^*-ax^*-by^*-1)^2)}}{(1+a(1-m)x^*+by^*)}, \\ \gamma _{02}{} & {} =\frac{(-1+m)(amx^{*2}-ax^{*2}-x^*)b}{((amx^*-ax^*-by^*-1)^2(1+a(1-m)x^*+by^*))}, \\ \gamma _{30}{} & {} =\frac{(-1+m)(abmy^{*2}-aby^{*2}+amy^*-ay^*)a(1-m)}{((amx^*-ax^*-by^*-1)^3 (1+a(1-m)x^*+by^*))}-\frac{E_1^2 l_1 p_1^2}{(E_1 l_1+x^* p_1)^4},\\ \gamma _{21}{} & {} =\frac{\frac{((-1+m)(-2abmx^*y^*+2aby^*-amx^*+ax^*+by^*+1)a(1-m)}{(amx^*-ax^*-by^*-1)^3}+\frac{(-1+m)(abmy^{*2}-aby^{*2}+amy^*-ay^*)b}{(amx^*-ax^*-by^*-1)^3)}}{(1+a(1-m)x^*+by^*)}, \\ \gamma _{12}{} & {} =\frac{\frac{(-1+m)(abmx^{*2}-abx^{*2}-bx^*)a(1-m)}{(amx^*-ax^*-by^*-1)^3}+\frac{(-1+m)(-2abmx^*y^*+2abx^*y^*-amx^*+ax^*+by^*+1)b}{(amx^*-ax^*-by^*-1)^3}}{(1+a(1-m)x^*+by^*)}, \\ \gamma _{03}{} & {} =\frac{(-1+m)(abmx^{*2}-abx^{*2}-bx^*)b}{((amx^*-ax^*-by^*-1)^3(1+a(1-m)x^*+by^*))},\\ \beta _{10}{} & {} =A_{21}=\frac{c(1-m)y^*(1+by^*)}{(1+a(1-m)x^*+by^*)^2}, \beta _{01}\\{} & {} =-d+\frac{c(1-m)x^*[1+a(1-m)x^*]}{(1+a(1-m)x^*+by^*)^2}-\frac{E_2^2 l_2}{(E_2l_2+p_2y^*)^2},\\ \beta _{20}{} & {} =-\frac{(-1+m)c(-by^{*2}-y^*)a(1-m)}{(amx^*-ax^*-by^*-1)^2 (1+a(1-m)x^*+by^*))}, \\ \beta _{11}{} & {} =\frac{\frac{(c(1-m)-(-1+m)c(amx^{*2}-ax^{*2}-x^*)a(1-m)}{(amx^*-ax^*-by^*-1)^2}-\frac{(-1+m)c(-by^{*2}-y^*)b}{(amx^*-ax^*-by^*-1)^2)}}{(1+a(1-m)x^*+by^*)},\\ \beta _{02}{} & {} =-\frac{(-1+m)c(amx^{*2}-ax^{*2}-x^*)b}{((amx^*-ax^*-by^*-1)^2(1+a(1-m)x^*+by^*))}\\{} & {} +\frac{(E_2+\epsilon _2)^2 l_2 p_2}{((E_2 l_2+\epsilon _2 l_2+p_2y^*)^2(l_2 (E_2+\epsilon _2)+p_2 y^*))}, \\ \beta _{30}{} & {} =-\frac{(-1+m)c(abmy^{*2}-aby^{*2}+amy^*-ay^*)a(1-m)}{((amx^*-ax^*-by^*-1)^3(1+a(1-m)x^*+by^*))}, \\ \beta _{21}{} & {} =-\frac{\frac{(-1+m)c(-2abmx^*y^*+2abx^* y^*-amx^*+ax^*+by^*+1)a(1-m)}{(amx^*-ax^*-by^*-1)^3}+\frac{(-1+m)c(abmy^{*2}-aby^{*2}+amy^*-ay^*)b}{(amx^*-ax^*-by^*-1)^3}}{(1+a(1-m)x^*+by^*)}, \\ \beta _{12}{} & {} =-\frac{\frac{(-1+m)c(abmx^{*2}-abx^{*2}-bx^*)a(1-m)}{(amx^*-ax^*-by^*-1)^3}+\frac{(-1+m)c(-2abmx^*y^*+2abx^*y^*-amx^*+ax^*+by^*+1)b}{(amx^*-ax^*-by^*-1)^3}}{(1+a(1-m)x^*+by^*)}, \\ \beta _{03}{} & {} =-\frac{(-1+m)c(abmx^{*2}-abx^{*2}-bx^*)b}{((amx^*-ax^*-by^*-1)^3(1+a(1-m)x^*+by^*))}\\{} & {} -\frac{(E_2+\epsilon _2)^2 l_2 p_2^2}{((E_2 l_2+\epsilon _2 l_2+p_2y^*)^3(l_2(E_2+\epsilon _2)+p_2y^*))}.\\ \end{aligned}$$

Appendix IV

$$\begin{aligned} a(\epsilon ){} & {} =-2x^*+1-\frac{((1-m)y^*-\frac{(-1+m)x^*y^*a(1-m)}{(amx^*-ax^*-by^*-1))}}{(1+a(1-m)x^*+by^*)}\\{} & {} -\frac{((E_1+e_2)-\frac{(E_1+e_2)x^*p_1}{(E_1 e_1+E_1 l_1+e_1 e_2+e_2 l_1+x^*p_1))}}{((l_1+e_1)(E_1+e_2)+p_1 x^*)},\\ b(\epsilon ){} & {} =-\frac{((1-m)x^*-\frac{(-1+m)x^*y^*b}{(amx^*-ax^*-by^*-1))}}{(1+a(1-m)x^*+by^*)},\\ c(\epsilon ){} & {} =\frac{(c(1-m)y^*-\frac{c(-1+m)x^*y^*a(1-m)}{(amx^*-ax^*-by^*-1))}}{(1+a(1-m)x^*+by^*)},\\ d(\epsilon ){} & {} =-d+\frac{(c(1-m)x^*-\frac{c(-1+m)x^*y^*b}{(amx^*-ax^*-by^*-1))}}{(1+a(1-m)x^*+by^*)}\\{} & {} -\frac{(E_2-\frac{E_2y^*p_2}{(E_2 l_2+p_2y^*))}}{(E_2 l_2+p_2y^*)},\\ p_{00}(\epsilon ){} & {} =x^*(-x^*+1)-\frac{(1-m)x^*y^*}{(1+a(1-m)x^*+by^*)}-\frac{(E_1+e_2)x^*}{((l_1+e_1)(E_1+\epsilon _2)+p_1x^*)},\\ p_{20}(\epsilon ){} & {} =-1+\frac{(-1+m)(-by^{*2}-y^*)a(1-m)}{((amx^*-ax^*-by^*-1)^2 (1+a(1-m)x^*+by^*))}\\{} & {} +\frac{(E_1+e_2)(E_1e_1+E_1 l_1+e_1 e_2+e_2 l_1)p_1}{((E_1 e_1+E_1 l_1+e_1 e_2+e_2 l_1+x^*p_1)^2 ((l_1+e_1)(E_1+e_2)+p_1 x^*))},\\ p_{11}(\epsilon ){} & {} =-\frac{(1-m-\frac{(-1+m)(amx^{*2}-ax^{*2}-x^*)a(1-m)}{(amx^*-ax^*-by^*-1)^2}-\frac{(-1+m)(-by^{*2}-y^*)b}{(amx^*-ax^*-by^*-1)^2)}}{(1+a(1-m)x^*+by^*)},\\ p_{02}(\epsilon ){} & {} =\frac{(-1+m)(amx^{*2}-ax^{*2}-x^*)b}{((amx^*-ax^*-by^*-1)^2 (1+a(1-m)x^*+by^*))},\\ q_{00}(\epsilon ){} & {} =-dy^*+\frac{c(1-m)x^*y^*}{(1+a(1-m)x^*+by^*)}-\frac{E_2y^*}{(E_2 l_2+p_2y^*)},\\ q_{20}(\epsilon ){} & {} =-\frac{(-1+m)c(-by^{*2}-y^*)a(1-m)}{((amx^*-ax^*-by^*-1)^2(1+a(1-m)x^*+by^*))},\\ q_{11}(\epsilon ){} & {} =\frac{(c(1-m)-\frac{(-1+m)c(amx^{*2}-ax^{*2}-x^*)a(1-m)}{(amx^*-ax^*-by^*-1)^2}-\frac{(-1+m)c(-by^{*2}-y^*)b}{(amx^*-ax^*-by^*-1)^2)}}{(1+a(1-m)x^*+by^*)},\\ q_{02}(\epsilon ){} & {} =-\frac{(-1+m)c(amx^{*2}-ax^{*2}-x^*)b}{((amx^*-ax^*-by^*-1)^2(1+a(1-m)x^*+by^*))}+\frac{E_2^2 l_2 p_2}{(E_2 l_2+p_2y^*)^3}.\\ \end{aligned}$$

Appendix V Expressions of \(k_{ij}\) and \(l_{ij}\)

$$\begin{aligned} k_{00}(\epsilon ){} & {} =\frac{p_{00}(\epsilon )}{\eta _{12}} -\eta _{11} \frac{p_{00}(\epsilon )}{\eta _{12}} -\eta _{00}(\epsilon ),\\ k_{10}(\epsilon ){} & {} =\frac{a(\epsilon ) \eta _{12} -b(\epsilon ) \eta _{11}}{\eta _{12}}-\eta _{11} \lbrace \frac{a(\epsilon ) \eta _{12} -b(\epsilon ) \eta _{11}}{\eta _{12}}+\frac{c(\epsilon ) \eta _{12} -d(\epsilon ) \eta _{11}}{\eta _{11}} \rbrace ,\\ k_{01}(\epsilon ){} & {} =\frac{a(\epsilon ) \eta _{12} +b(\epsilon ) (1-\eta _{11})}{\eta _{12}}-\eta _{11} \lbrace \\{} & {} \frac{a(\epsilon ) \eta _{12} +b(\epsilon )(1-\eta _{11})}{\eta _{12}}+\frac{c(\epsilon ) \eta _{12} +d(\epsilon ) (1-\eta _{11})}{\eta _{11}} \rbrace -1,\\ k_{20}(\epsilon ){} & {} =2\frac{p_{20}(\epsilon ) \eta _{12}^{2}-p_{11}(\epsilon ) \eta _{12} \eta _{11}+p_{02}(\epsilon ) \eta _{11}^{2}}{\eta _{12}}\\{} & {} -2\eta _{11} \frac{p_{20}(\epsilon ) \eta _{12}^{2}-p_{11}(\epsilon ) \eta _{12} \eta _{11}+p_{02}(\epsilon ) \eta _{11}^{2}}{\eta _{12}}+2q_{11}(\epsilon ) \eta _{12} \eta _{11}-2q_{02}(\epsilon ) \eta _{11}^{2},\\ k_{11}(\epsilon ){} & {} =\frac{2p_{20}(\epsilon ) \eta _{12}^{2}-p_{11}(\epsilon ) \eta _{12} \eta _{11}+p_{11}(\epsilon ) \eta _{12} (1-\eta _{11})-2p_{02}(\epsilon ) \eta _{11} (1-\eta _{11})}{\eta _{12}} \\{} & {} -\eta _{11} \frac{2p_{20}(\epsilon ) \eta _{12}^{2}-p_{11}(\epsilon ) \eta _{12} \eta _{11}+p_{11}(\epsilon ) \eta _{12} (1-\eta _{11})-2p_{02}(\epsilon ) \eta _{11} (1-\eta _{11})}{\eta _{12}}\\{} & {} -q_{11}(\epsilon ) \eta _{12} (1-\eta _{11})+q_{11}(\epsilon ) \eta _{11} \eta _{12}+2q_{02}(\epsilon ) \eta _{11} (1-\eta _{11}),\\ k_{02}(\epsilon ){} & {} =2\frac{p_{20}(\epsilon ) \eta _{12}^{2}+p_{11}(\epsilon ) \eta _{12} (1-\eta _{11})+p_{02}(\epsilon ) (1-\eta _{11})^{2}}{\eta _{12}}\\{} & {} -2\eta _{11} \frac{p_{20}(\epsilon ) \eta _{12}^{2}+p_{11}(\epsilon ) (1-\eta _{11})+p_{02}(\epsilon ) (1-\eta _{11})^{2}}{\eta _{12}}-2q_{11}(\epsilon ) \eta _{12} (1-\eta _{11})\\{} & {} -2q_{02}(\epsilon ) (1-\eta _{11})^{2},\\ l_{00}(\epsilon ){} & {} = \frac{ \eta _{11}p_{00}(\epsilon )}{\eta _{12}}+q_{00}(\epsilon ),\\ l_{10}(\epsilon ){} & {} =\eta _{11} \left\{ \frac{a(\epsilon ) \eta _{12} -b(\epsilon ) \eta _{11}}{\eta _{12}}+\frac{c(\epsilon ) \eta _{12} -d(\epsilon ) \eta _{11}}{\eta _{11}} \right\} ,\\ l_{01}(\epsilon ){} & {} =\eta _{11} \left\{ \frac{a(\epsilon ) \eta _{12} +b(\epsilon )(1-\eta _{11})}{\eta _{12}}+\frac{c(\epsilon ) \eta _{12} +d(\epsilon ) (1-\eta _{11})}{\eta _{11}} \right\} ,\\ l_{20}(\epsilon ){} & {} =2\eta _{11} \frac{p_{20}(\epsilon ) \eta _{12}^{2}-p_{11}(\epsilon ) \eta _{12} \eta _{11}+p_{02}(\epsilon ) \eta _{11}^{2}}{\eta _{12}}-2q_{11}(\epsilon ) \eta _{12} \eta _{11}+2q_{02}(\epsilon ) \eta _{11}^{2},\\ l_{11}(\epsilon ){} & {} =\eta _{11} \frac{2p_{20}(\epsilon ) \eta _{12}^{2}-p_{11}(\epsilon ) \eta _{12} \eta _{11}+p_{11}(\epsilon ) \eta _{12} (1-\eta _{11})-2p_{02}(\epsilon ) \eta _{11} (1-\eta _{11})}{\eta _{12}}\\{} & {} +q_{11}(\epsilon ) \eta _{12} (1-\eta _{11})\\{} & {} -q_{11}(\eta ) \eta _{11} \eta _{12}-2q_{02}(\epsilon ) \eta _{11} (1-\eta _{11}),\\ l_{02}(\epsilon ){} & {} =2 \eta _{11} \frac{p_{20}(\epsilon ) \eta _{12}^{2}+p_{11}(\epsilon ) \eta _{12} (1-\eta _{11})+p_{02}(\epsilon ) (1-\eta _{11})^{2}}{\eta _{12}}\\{} & {} +2q_{11}(\epsilon ) \eta _{12} (1-\eta _{11})+2q_{02}(\epsilon ) (1-\eta _{11})^{2}.\\ \end{aligned}$$

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Majumdar, P., Ghosh, U., Sarkar, S. et al. Study of co-dimension two bifurcation of a prey–predator model with prey refuge and non-linear harvesting on both species. Rend. Circ. Mat. Palermo, II. Ser 72, 4067–4100 (2023). https://doi.org/10.1007/s12215-023-00881-9

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