Effect of demographic stochasticity in the persistence zone of a two-patch model with nonlinear harvesting

Document Type : Research Article

Authors

• D. Mondal
• M. Sen
• P.S. Mandal

Department of Mathematics, National Institute of Technology Patna, Patna, Bihar, India.

10.22067/ijnao.2024.88668.1463

Abstract

In this study, Allee type, single-species (prey), two-patch model with nonlinear harvesting rate, and species migration across two patches have been developed and analyzed. As we all know, the population of any species in an ecosystem is greatly dependent on the carrying capacity of the corre-sponding ecosystem; the main focus of our work is on how carrying capacity affects system dynamics in the presence and absence of randomness (de-terministic and stochastic case, respectively). In the deterministic case, we find that the carrying capacity of both patches increases the number of interior equilibrium points, and a maximum of eight interior equilib-rium points can be observed. Also, we observe some interesting dynamics, including bi-stability, tri-stability, and catastrophic bifurcations. On the other hand, we use the continuous-time Markov chain modeling approach to construct an equivalent stochastic model of the corresponding determin-istic model based on deterministic assumptions. Based on the extinction or persistence of the species, we compare the dynamics of deterministic and stochastic models in order to assess the impact of demographic stochas-ticity on the population of the species in two patches. The stochastic model shows the possibility of species extinction in a finite amount of time, whereas the deterministic model shows the persistence of the species at the same time, which is the major difference between these two models. We also derive the implicit equation for the expected time needed for species extinction. Finally, a graphic is used to illustrate how the patch’s carrying capacity affects the expected time.

Keywords

• Stability analysis
• Bifurcation
• Continuous-time Markov chain (CTMC)
• Expected time to extinction

Main Subjects

• Modeling and Simulation

---

References

[1] Akhi, A.A., Kamrujjaman, M., Nipa, K.F., and Khan, T. A continuous-time Markov chain and stochastic differential equations approach for modeling malaria propagation, Healthcare Analytics 4 (2023), 100239.
[2] Allen, L.J. An introduction to stochastic processes with applications to biology, CRC press, 2010.
[3] Allen, L.J. A primer on stochastic epidemic models: Formulation, nu-merical simulation, and analysis, Infect. Dis. Model. 2 (2) (2017), 128–142.
[4] Allen, L.J., and Lahodny Jr, G.E. Extinction thresholds in deterministic and stochastic epidemic models, J. Biol. Dyn. 6 (2) (2012), 590–611.
[5] Allen, L.J., and vanden Driessche, P. Relations between determinis-tic and stochastic thresholds for disease extinction in continuous-and discrete-time infectious disease models, Math. Biosci. 243, 1 (2013), 99–108.
[6] Bhowmick, A.R., Bandyopadhyay, S., Rana, S., and Bhattacharya, S. A simple approximation of moments of the quasi-equilibrium distribution of an extended stochastic theta-logistic model with non-integer powers, Math. Biosci. 271 (2016), 96–112.
[7] Crépin, A.-S., Biggs, R., Polasky, S., Troell, M., and DeZeeuw, A. Regime shifts and management, Ecol. Econ. 84 (2012), 15–22.
[8] Ditlevsen, S., and Samson, A. Introduction to stochastic models in bi-ology, Stochastic Biomathematical Models: With Applications to Neu-ronal Modeling, (2013) 3–35.
[9] Gillespie, D.T. A general method for numerically simulating the stochas-tic time evolution of coupled chemical reactions, J. Comput. Phys. 22 (4) (1976), 403–434.
[10] Hale, J.K. Functional differential equations, In Analytic Theory of Differential Equations: The Proceedings of the Conference at Western Michigan University, Kalamazoo, from 30 April to 2 May 1970 (2006), Springer, 9–22.
[11] Hilker, F.M., Langlais, M., and Malchow, H. The Allee effect and in-fectious diseases: extinction, multistability, and the (dis-) appearance of oscillations, Am. Nat. 173 (1) (2009), 72–88.
[12] Jang, S. R.-J., and Baglama, J. Continuous-time predator–prey models with parasites, J. Biol. Dyn. 3 (1) (2009), 87–98.
[13] Maity, S., and Mandal, P.S. A comparison of deterministic and stochas-tic plant-vector-virus models based on probability of disease extinction and outbreak, Bull. Math. Biol. 84 (3) (2022), 41.
[14] Maity, S., and Mandal, P.S. The effect of demographic stochasticity on Zika virus transmission dynamics: Probability of disease extinction, sensitivity analysis, and mean first passage time, Chaos: An Interdis-ciplinary Journal of Nonlinear Science 34 (3) (2024).
[15] Maliyoni, M. Probability of disease extinction or outbreak in a stochastic epidemic model for west Nile virus dynamics in birds, Acta Biotheor. 69 (2) (2021), 91–116.
[16] Maliyoni, M., Chirove, F., Gaff, H.D., and Govinder, K.S. A stochastic tick-borne disease model: Exploring the probability of pathogen persis-tence, Bull. Math. Biol. 79 (2017), 1999–2021.
[17] Maliyoni, M., Chirove, F., Gaff, H.D., and Govinder, K.S. A stochas-tic epidemic model for the dynamics of two pathogens in a single tick population, Theor. Popul. Biol. 127 (2019), 75–90.
[18] Mandal, P.S., Allen, L.J., and Banerjee, M. Stochastic modeling of phy-toplankton allelopathy, Appl. Math. Model. 38, (5-6) (2014), 1583–1596.
[19] Mandal, P.S., Kumar, U., Garain, K., and Sharma, R. Allee effect can simplify the dynamics of a prey-predator model, J. Appl. Math. Comput. 63 (2020), 739–770.

[20] Mandal, P.S., and Maity, S. Impact of demographic variability on the disease dynamics for honeybee model, Chaos: An Interdisciplinary Journal of Nonlinear Science 32 (8) (2022), 083120.
[21] Nandi, A., and Allen, L.J. Stochastic multigroup epidemic models: Du-ration and final size, Modeling, Stochastic Control, Optimization, and Applications (2019), 483–507.
[22] Ndii, M.Z., and Supriatna, A.K. Stochastic mathematical models in epidemiology, Information 20 (2017), 6185–6196.
[23] Nipa, K.F., Jang, S. R.-J., and Allen, L.J. The effect of demographic and environmental variability on disease outbreak for a dengue model with a seasonally varying vector population, Math. Biosci. 331 (2021), 108516.
[24] Perko, L. Differential equations and dynamical systems, vol.7. Springer Science & Business Media, 2013.
[25] Polovina, J.J. Climate variation, regime shifts, and implications for sustainable fisheries, Bull. Marine Sci.76 (2) (2005), 233–244.
[26] Saha, B., Bhowmick, A.R., Chattopadhyay, J., and Bhattacharya, S. On the evidence of an Allee effect in herring populations and consequences for population survival: A model-based study, Ecol. Model. 250 (2013), 72–80.
[27] Sau, A., Bhattacharya, S., and Saha, B. Recognizing and prevention of probable regime shift in density regulated and Allee type stochastic har-vesting model with application to herring conservation, arXiv preprint arXiv:2108.07534 (2021).
[28] Sau, A., Saha, B., and Bhattacharya, S. An extended stochastic Allee model with harvesting and the risk of extinction of the herring population, J. Theor. Biol. 503 (2020), 110375.
[29] Scheffer, M., Carpenter, S., Foley, J.A., Folke, C., and Walker, B. Catas-trophic shifts in ecosystems, Nature 413, 6856 (2001), 591–596.

[30] Scheffer, M., and Carpenter, S.R. Catastrophic regime shifts in ecosys-tems: linking theory to observation, Trend Ecol. Evol. 18 (12) (2003), 648–656.
[31] Sen, M., and Banerjee, M. Rich global dynamics in a prey–predator model with Allee effect and density dependent death rate of predator, Inter. J. Bifurcat.Chaos 25, 03 (2015), 1530007.
[32] Sibly, R.M., Barker, D., Denham, M.C., Hone, J., and Pagel, M. On the regulation of populations of mammals, birds, fish, and insects, Science 309 (5734) (2005), 607–610.
[33] Stephano, M.A., Irunde, J.I., Mwasunda, J.A., and Chacha, C.S. A con-tinuous time Markov chain model for the dynamics of bovine tuberculosis in humans and cattle, Ricerche di Matematica (2022), 1–27.
[34] Swift, R.J. A stochastic predator-prey model, Irish Math. Soc. Bull. 48, 57-63 (2002), 646.
[35] Vishwakarma, K., and Sen, M. Role of Allee effect in prey and hunting cooperation in a generalist predator, Math. Comput. Simul. 190 (2021), 622–640.