# American Institute of Mathematical Sciences

doi: 10.3934/naco.2021035
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## Modelling and analysis of prey-predator model involving predation of mature prey using delay differential equations

 1 Department of Mathematics, Lovely Professional University, Phagwara, Punjab -144411, INDIA 2 Ph.D. Scholar, Lovely Professional University, Phagwara, Punjab -144411, INDIA

* Corresponding author: Shiv Raj

This paper is handled by V. Sree Hari Rao as the guest editor

Received  December 2020 Revised  August 2021 Early access September 2021

In this paper, the modelling and analysis of prey-predator model involving predation of mature prey is done using DDE. Equilibrium points are calculated and stability analysis is performed about non-zero equilibrium point. Delay parameter destabilizes the system and triggers asymptotic stability when value of delay parameter is below the critical point. Hopf bifurcation is observed when the value of delay parameter crosses the critical point. Sensitivity analysis has also been performed to look into the effect of other parameters on the state variables. The numerical results are substantiated using MATLAB.

Citation: Pankaj Kumar, Shiv Raj. Modelling and analysis of prey-predator model involving predation of mature prey using delay differential equations. Numerical Algebra, Control and Optimization, doi: 10.3934/naco.2021035
##### References:
 [1] R. Arditi and L. R. Ginzburg, Coupling in predator-prey dynamics: Ratio dependence, Journal of Theoretical Biology, 139 (1989), 311-326. [2] O. Bernard and S. Souissi, Qualitative behavior of stage structured populations: application to structural validation, Journal of Mathematical Biology, 37 (1998), 291-308.  doi: 10.1007/s002850050130. [3] J. Chattopadhyay and A. Ovide, A predator-prey model with disease in the prey, Nonlinear Analysis: Theory, Methods & Applications, 36 (1999), 747-766.  doi: 10.1016/S0362-546X(98)00126-6. [4] F. Chen, Z. Ma and H. Zhang, Global asymptotical stability of the positive equilibrium of the Lotka-Volterra prey-predator model incorporating a constant number of prey refuge, Nonlinear Analysis: Real World Applications, 13 (2012), 2790-2793.  doi: 10.1016/j.nonrwa.2012.04.006. [5] J. Cui, L. Chen and W. Wang, The effect of dispersal on population growth with stage- structure, Computers and Mathematics with Applications, 39 (2000), 91-102.  doi: 10.1016/S0898-1221(99)00316-8. [6] J. Cui and and Y. Takeuchi, A predator-prey system with a stage structure for the prey, Mathematical and Computer Modelling, 44 (2006), 1126-1132.  doi: 10.1016/j.mcm.2006.04.001. [7] F. M. D. Gulland, The impact of infectious diseases on wild animal populations: A review, Ecology of infectious diseases in natural populations, 1 (1995), 20-50. [8] C. S. Holling, The functional response of predators to prey density and its role in mimicry and population regulation, The Memoirs of the Entomological Society of Canada, 97 (1973), 5-60. [9] C. Li, Dynamics of a network-based SIS epidemic model with nonmonotone incidence rate, Physica A, 427 (2005), 234-243.  doi: 10.1016/j.physa.2015.02.023. [10] Alfred J. Lotka, Analytical note on certain rhythmic relations in organic systems, Proceedings of the National Academy of Sciences, 6 (1920), 410-415.  doi: 10.1073/pnas.6.7.410. [11] R. M. May, Limit cycles in prey-predator communities, Science, 177 (1972), 900-902. [12] R. M. May, Stability and Complexity in Model Ecosystems, Princeton U. P., 1973. [13] P. Pal, M. Haque and P. Mandal, Dynamics of a predator-prey model with disease in the predator, Math. Meth. Appl. Sci., 37 (2014), 2429-2450.  doi: 10.1002/mma.2988. [14] S. Ruan, Absolute stability, conditional stability and bifurcation in Kolmogorov-type predator-prey system with discrete delays, Quart. Appl. Math., 59 (2001), 159-173.  doi: 10.1090/qam/1811101. [15] S. Ruan and J. Wei, On the zeros of transcendental functions with applications to stability of delay differential equations, Dynam. Contin. Discr. Impus.Syst., 10 (2003), 863-874. [16] S. Ruan, On nonlinear dynamics of predator-prey models with discrete delays, Math. Model. Nat. Phenom, 4 (2009), 140-188.  doi: 10.1051/mmnp/20094207. [17] Th éodore Vogel, Dynamique théorique et heredité, Rend. Sem. Mat. Univ. Politec. Torino, 21 (1961), 87-98. [18] V. Volterra, Variations and fluctuations of the numbers of individuals in coexisting animal populations, Mem. R. Comitato Talassogr. Ital. Mem, 131 (1927). [19] R. Xu and S. Zhang, Modelling and analysis of a delayed predator-prey model with disease in the predator, Appl. Math. Comput., 224 (2013), 372-386.  doi: 10.1016/j.amc.2013.08.067. [20] X. Zhang, X. Chen and A. U. Neumann, The stage-structured predator-prey model and optimal harvesting policy, Mathematical Biosciences, 168 (2000), 201-210.  doi: 10.1016/S0025-5564(00)00033-X.

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##### References:
 [1] R. Arditi and L. R. Ginzburg, Coupling in predator-prey dynamics: Ratio dependence, Journal of Theoretical Biology, 139 (1989), 311-326. [2] O. Bernard and S. Souissi, Qualitative behavior of stage structured populations: application to structural validation, Journal of Mathematical Biology, 37 (1998), 291-308.  doi: 10.1007/s002850050130. [3] J. Chattopadhyay and A. Ovide, A predator-prey model with disease in the prey, Nonlinear Analysis: Theory, Methods & Applications, 36 (1999), 747-766.  doi: 10.1016/S0362-546X(98)00126-6. [4] F. Chen, Z. Ma and H. Zhang, Global asymptotical stability of the positive equilibrium of the Lotka-Volterra prey-predator model incorporating a constant number of prey refuge, Nonlinear Analysis: Real World Applications, 13 (2012), 2790-2793.  doi: 10.1016/j.nonrwa.2012.04.006. [5] J. Cui, L. Chen and W. Wang, The effect of dispersal on population growth with stage- structure, Computers and Mathematics with Applications, 39 (2000), 91-102.  doi: 10.1016/S0898-1221(99)00316-8. [6] J. Cui and and Y. Takeuchi, A predator-prey system with a stage structure for the prey, Mathematical and Computer Modelling, 44 (2006), 1126-1132.  doi: 10.1016/j.mcm.2006.04.001. [7] F. M. D. Gulland, The impact of infectious diseases on wild animal populations: A review, Ecology of infectious diseases in natural populations, 1 (1995), 20-50. [8] C. S. Holling, The functional response of predators to prey density and its role in mimicry and population regulation, The Memoirs of the Entomological Society of Canada, 97 (1973), 5-60. [9] C. Li, Dynamics of a network-based SIS epidemic model with nonmonotone incidence rate, Physica A, 427 (2005), 234-243.  doi: 10.1016/j.physa.2015.02.023. [10] Alfred J. Lotka, Analytical note on certain rhythmic relations in organic systems, Proceedings of the National Academy of Sciences, 6 (1920), 410-415.  doi: 10.1073/pnas.6.7.410. [11] R. M. May, Limit cycles in prey-predator communities, Science, 177 (1972), 900-902. [12] R. M. May, Stability and Complexity in Model Ecosystems, Princeton U. P., 1973. [13] P. Pal, M. Haque and P. Mandal, Dynamics of a predator-prey model with disease in the predator, Math. Meth. Appl. Sci., 37 (2014), 2429-2450.  doi: 10.1002/mma.2988. [14] S. Ruan, Absolute stability, conditional stability and bifurcation in Kolmogorov-type predator-prey system with discrete delays, Quart. Appl. Math., 59 (2001), 159-173.  doi: 10.1090/qam/1811101. [15] S. Ruan and J. Wei, On the zeros of transcendental functions with applications to stability of delay differential equations, Dynam. Contin. Discr. Impus.Syst., 10 (2003), 863-874. [16] S. Ruan, On nonlinear dynamics of predator-prey models with discrete delays, Math. Model. Nat. Phenom, 4 (2009), 140-188.  doi: 10.1051/mmnp/20094207. [17] Th éodore Vogel, Dynamique théorique et heredité, Rend. Sem. Mat. Univ. Politec. Torino, 21 (1961), 87-98. [18] V. Volterra, Variations and fluctuations of the numbers of individuals in coexisting animal populations, Mem. R. Comitato Talassogr. Ital. Mem, 131 (1927). [19] R. Xu and S. Zhang, Modelling and analysis of a delayed predator-prey model with disease in the predator, Appl. Math. Comput., 224 (2013), 372-386.  doi: 10.1016/j.amc.2013.08.067. [20] X. Zhang, X. Chen and A. U. Neumann, The stage-structured predator-prey model and optimal harvesting policy, Mathematical Biosciences, 168 (2000), 201-210.  doi: 10.1016/S0025-5564(00)00033-X.
The equilibrium $E^*$($P_r^*$, $P_d^*$) is stable in the absence of delay i.e. when $\tau$ = 0
The equilibrium $E^*$($P_r^*$, $P_d^*$) is asymptotically stable when delay is less than critical value i.e. when $\tau$ $<$ 8.23
Phase plane of equilibrium $E^*$($P_r^*$, $P_d^*$) when the delay is less than critical value i.e. $\tau$ $<$ 8.23
The equilibrium $E^*$($P_r^*$, $P_d^*$) losses stability and shows Hopf-bifurcation when delay is crosses the critical value i.e. when $\tau$ $\geq$ 8.23
Phase plane of equilibrium $E^*$($P_r^*$, $P_d^*$) when the delay crosses the critical value i.e. $\tau$ $\geq$ 8.23
Time series graph between partial changes in $P_d$ (predator populations) for different values of the parameter $\alpha$
Time series graph between partial changes in $P_r$ (Prey populations) for different values of the parameter $\delta$
Description of the parameters of the system (1) – (2)
 Parameter Discription b Intrinsic birth rate of prey population $\alpha$ Intra specific competition rate among prey $\beta$ Inter specific competition rate d Death rate of predator population $\gamma$ Inter specific competition rate $\delta$ Intra specific competition rate among predator $\tau$ Delay parameter
 Parameter Discription b Intrinsic birth rate of prey population $\alpha$ Intra specific competition rate among prey $\beta$ Inter specific competition rate d Death rate of predator population $\gamma$ Inter specific competition rate $\delta$ Intra specific competition rate among predator $\tau$ Delay parameter
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