# American Institute of Mathematical Sciences

September  2021, 11(3): 391-405. doi: 10.3934/naco.2020033

## Dynamics of a stage structure prey-predator model with ratio-dependent functional response and anti-predator behavior of adult prey

 1 Department of Applied Science, Haldia Institute of Technology, Haldia-721657, W. B., India 2 Department of Mathematics, Ramsaday College, Amta, Howrah, India 3 Department of Applied Mathematics with Oceanology, and Computer Programming, Vidyasagar University, Midnapore -721102, W. B., India

* Corresponding author: Prabir Panja, prabirpanja@gmail.com

Received  February 2020 Revised  February 2020 Published  September 2021 Early access  May 2020

In this paper, a predator-prey interaction model among juvenile prey, adult prey and predator has been developed where stage structure is considered on prey species. The functional responses has been considered as ratio dependent. It is assumed that that the adult prey is strong enough such that it has an anti-predator characteristic. Global dynamics of the co-existing equilibrium point has been discussed with the help of the geometric approach. Furthermore, it is established that the proposed system undergoes through a Hopf bifurcation with respect to some important parameters. Finally, some numerical simulations have been done to test our theoretical results.

Citation: Prabir Panja, Soovoojeet Jana, Shyamal kumar Mondal. Dynamics of a stage structure prey-predator model with ratio-dependent functional response and anti-predator behavior of adult prey. Numerical Algebra, Control and Optimization, 2021, 11 (3) : 391-405. doi: 10.3934/naco.2020033
##### References:
 [1] M. Andersson and C. G. Wiklund, Clumping versus spacing out: Experiments on nest predation in fieldfares (Turdus pilaris), Animal Behavior, 26 (1978), 1207-1212. [2] R. Arditi and L. R. Ginzburg, Coupling in predator-prey dynamics ratio dependence, Journal of Theoretical Biology, 139 (1989), 311-326. [3] O. Arino, A. Abdllaoui, J. Mikram and J. Chattopadhyay, Infection on prey population may act as biological control in ratio-dependent predator-prey model, Nonlinearity, 17 (2004), 1101-1116.  doi: 10.1088/0951-7715/17/3/018. [4] M. Bandyopadhyay and J. Chattopadhyay, Ratio-dependent predator-prey model: Effect of environmental fluctuation and stability, Nonlinearity, 18 (2005), 913-936.  doi: 10.1088/0951-7715/18/2/022. [5] M. Banerjee, Self-replication of spatial patterns in a ratio-dependent predator-prey model, Mathematical and Computer Modelling, 51 (2010), 44-52.  doi: 10.1016/j.mcm.2009.07.015. [6] A. A. Berryman, The origin and evolution of predator-prey theory, Ecology, 73 (1992), 1530-1535. [7] G. Birkoff, G. C. Rota, Ordinary Differential Equations, Ginn, Boston, 1982. [8] Y. Choh, M. Lgnacio, M. W. Sabelis and A. Janssen, Predator-prey role reversale, juvenile experience and adult antipredator behavior, Scientific Reports, 2 (2012), 7-28. [9] Y. H. Fan and W. T. Li, Permanence for a delayed discrete ratio-dependent predator-prey system with Holling type functional response, Journal of Mathematical Analysis and Applications, 299 (2004), 357-374.  doi: 10.1016/j.jmaa.2004.02.061. [10] H. I. Freedman and A. M. Mathsen, Persistence in predator-prey systems with ratio-dependent predator influence, Bulletin of Mathematical Biology, 55 (1993), 817-827. [11] S. Gakkhar and K. Negi, A mathematical model for viral infection in toxin producing phytoplankton and zooplankton system, Applied Mathematics and Computation, 179 (2006), 301-313.  doi: 10.1016/j.amc.2005.11.166. [12] A. Hugo, E. S. Massawe and O. D. Makinde, An ecoepidemiological mathematical model with treatment and disease infection in both prey and predator population, Journal of Ecology and The Natural Environment, 4 (2012), 266-279.  doi: 10.1016/j.physa.2013.07.077. [13] A. Janssen, F. Faraji, T. Van Der Hammen, S. Magalhaes and M. W. Sabelis, Interspecific infanticide deters predators, Ecology Letters, 5 (2002), 490-494. [14] Y. Kuang and E. Beretta, Global qualitative analysis of a ratio-dependent predator-prey system, Journal of Mathematical Biology, 43 (2001), 221-290.  doi: 10.1007/s002850100097. [15] M. Y. Li and J. S. Muldowney, A geometric approach global stability problems, SIAM Journal on Mathematical Analysis, 27 (1996), 1070-1083.  doi: 10.1137/S0036141094266449. [16] A. J. Lotka, Elements of Physical Biology, Williams and Wilkins, Baltimore, 1925. [17] P. Panja and S. K. Mondal, Stability analysis of coexistence of three species prey-predator model, Nonlinear Dynamics, 81 (2015), 373-382.  doi: 10.1007/s11071-015-1997-1. [18] Y. Pei, L. Chen, Q. Zhang and C. Li, Extinction and permanence of one-preymulti-predators of Holling type Ⅱ function response system with impulsive biological control, Journal of Theoretical Biology, 235 (2005), 495-503.  doi: 10.1016/j.jtbi.2005.02.003. [19] G. A. Polis, C. A. Myers and R. D. Holt, The ecology and evaluation of intraguild predation-potential competitors that eat each other, Annual Review of Ecology and Systematics, 20 (1989), 297-330. [20] S. Ruan, A. Ardito, P. Ricciardi and D. L. DeAngalis, Coexistence in competition models with density dependent mortality, Comptes Rendus Biologies, 330 (2007), 845-854. [21] C. B. Stanford, The influence of chimpanzee predation on group size and anti-predator behavior in red colobus monkeys, Animal Behavior, 49 (1995), 577-587. [22] B. Tang and Y. Xiao, Bifurcation analysis of a predator-prey model with anti-predator behavior, Chaos, Solitons & Fractals, 70 (2015), 58-68.  doi: 10.1016/j.chaos.2014.11.008. [23] F. Wei and Q. Fu, Hopf bifurcation and stability for predator-prey systems with Beddington-DeAngelis type functional response and stage structure for prey incorporating refuge, Applied Mathematical Modelling, 40 (2016), 126-134.  doi: 10.1016/j.apm.2015.04.042.

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##### References:
 [1] M. Andersson and C. G. Wiklund, Clumping versus spacing out: Experiments on nest predation in fieldfares (Turdus pilaris), Animal Behavior, 26 (1978), 1207-1212. [2] R. Arditi and L. R. Ginzburg, Coupling in predator-prey dynamics ratio dependence, Journal of Theoretical Biology, 139 (1989), 311-326. [3] O. Arino, A. Abdllaoui, J. Mikram and J. Chattopadhyay, Infection on prey population may act as biological control in ratio-dependent predator-prey model, Nonlinearity, 17 (2004), 1101-1116.  doi: 10.1088/0951-7715/17/3/018. [4] M. Bandyopadhyay and J. Chattopadhyay, Ratio-dependent predator-prey model: Effect of environmental fluctuation and stability, Nonlinearity, 18 (2005), 913-936.  doi: 10.1088/0951-7715/18/2/022. [5] M. Banerjee, Self-replication of spatial patterns in a ratio-dependent predator-prey model, Mathematical and Computer Modelling, 51 (2010), 44-52.  doi: 10.1016/j.mcm.2009.07.015. [6] A. A. Berryman, The origin and evolution of predator-prey theory, Ecology, 73 (1992), 1530-1535. [7] G. Birkoff, G. C. Rota, Ordinary Differential Equations, Ginn, Boston, 1982. [8] Y. Choh, M. Lgnacio, M. W. Sabelis and A. Janssen, Predator-prey role reversale, juvenile experience and adult antipredator behavior, Scientific Reports, 2 (2012), 7-28. [9] Y. H. Fan and W. T. Li, Permanence for a delayed discrete ratio-dependent predator-prey system with Holling type functional response, Journal of Mathematical Analysis and Applications, 299 (2004), 357-374.  doi: 10.1016/j.jmaa.2004.02.061. [10] H. I. Freedman and A. M. Mathsen, Persistence in predator-prey systems with ratio-dependent predator influence, Bulletin of Mathematical Biology, 55 (1993), 817-827. [11] S. Gakkhar and K. Negi, A mathematical model for viral infection in toxin producing phytoplankton and zooplankton system, Applied Mathematics and Computation, 179 (2006), 301-313.  doi: 10.1016/j.amc.2005.11.166. [12] A. Hugo, E. S. Massawe and O. D. Makinde, An ecoepidemiological mathematical model with treatment and disease infection in both prey and predator population, Journal of Ecology and The Natural Environment, 4 (2012), 266-279.  doi: 10.1016/j.physa.2013.07.077. [13] A. Janssen, F. Faraji, T. Van Der Hammen, S. Magalhaes and M. W. Sabelis, Interspecific infanticide deters predators, Ecology Letters, 5 (2002), 490-494. [14] Y. Kuang and E. Beretta, Global qualitative analysis of a ratio-dependent predator-prey system, Journal of Mathematical Biology, 43 (2001), 221-290.  doi: 10.1007/s002850100097. [15] M. Y. Li and J. S. Muldowney, A geometric approach global stability problems, SIAM Journal on Mathematical Analysis, 27 (1996), 1070-1083.  doi: 10.1137/S0036141094266449. [16] A. J. Lotka, Elements of Physical Biology, Williams and Wilkins, Baltimore, 1925. [17] P. Panja and S. K. Mondal, Stability analysis of coexistence of three species prey-predator model, Nonlinear Dynamics, 81 (2015), 373-382.  doi: 10.1007/s11071-015-1997-1. [18] Y. Pei, L. Chen, Q. Zhang and C. Li, Extinction and permanence of one-preymulti-predators of Holling type Ⅱ function response system with impulsive biological control, Journal of Theoretical Biology, 235 (2005), 495-503.  doi: 10.1016/j.jtbi.2005.02.003. [19] G. A. Polis, C. A. Myers and R. D. Holt, The ecology and evaluation of intraguild predation-potential competitors that eat each other, Annual Review of Ecology and Systematics, 20 (1989), 297-330. [20] S. Ruan, A. Ardito, P. Ricciardi and D. L. DeAngalis, Coexistence in competition models with density dependent mortality, Comptes Rendus Biologies, 330 (2007), 845-854. [21] C. B. Stanford, The influence of chimpanzee predation on group size and anti-predator behavior in red colobus monkeys, Animal Behavior, 49 (1995), 577-587. [22] B. Tang and Y. Xiao, Bifurcation analysis of a predator-prey model with anti-predator behavior, Chaos, Solitons & Fractals, 70 (2015), 58-68.  doi: 10.1016/j.chaos.2014.11.008. [23] F. Wei and Q. Fu, Hopf bifurcation and stability for predator-prey systems with Beddington-DeAngelis type functional response and stage structure for prey incorporating refuge, Applied Mathematical Modelling, 40 (2016), 126-134.  doi: 10.1016/j.apm.2015.04.042.
Local stability of the equilibrium point $E_1$
Local stability of the equilibrium point $E^{*}$
Global stability of interior equilibrium
Bifurcation diagram of system $(1)$ with respect to $\eta$
Bifurcation diagram of system $(1)$ with respect to $\beta$
Bifurcation diagram of system $(1)$ with respect to $\beta_1$
Bifurcation diagram of system $(1)$ with respect to $\beta_2$
Bifurcation diagram of system $(1)$ with respect to $\gamma$
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