Bifurcation analysis and optimal control in a predator-prey-infection model with additional food
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Abstract
We propose and analyze a three-dimensional eco-epidemiological model involving susceptible and infected prey and predators, in which the predators are supplemented with a constant externally supplied food supply. The model incorporates nonlinear disease transmission and predator feeding saturation through a generalized Holling type II functional response. We investigate the system's dynamics analytically and numerically by examining the existence and stability of equilibria, as well as Hopf, transcritical, and saddle-node bifurcations. One- and two-parameter bifurcation analyses reveal rich dynamics, including limit cycles, period doubling, and chaotic oscillations. Our findings indicate that disease transmission can destabilize the system, while the inclusion of additional food enhances stability and can suppress chaos.
Furthermore, we extend the model by introducing a time-dependent optimal control variable representing additional food supply, and derive an optimal strategy using Pontryagin's Maximum Principle. Numerical simulations show that optimal control effectively reduces disease prevalence and stabilizes population dynamics. This study highlights the potential of ecological interventions, such as strategic food supplementation, in regulating complex eco-epidemiological systems.
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References
P.A. Abrams and J.D. Roth, The effects of enrichment of three-species food chains with nonlinear functional responses, Ecology, 75(4) (1994), 1118–1130. https://doi.org/10.2307/1939435
P. Akhtar, S. Karmakar and G. Samanta, Exploring tri-trophic food chain model with toxicity and additional food: Unraveling bifurcation, chaos, density variation and bistability, Int. J. Biomath., 2550024 (2025). https://doi.org/10.1142/S179352452550024X
R.M. Anderson and R.M. May, Population biology of infectious diseases: Part I, Nature, 280(5721) (1979), 361–367. https://doi.org/10.1038/280361a0
R.M. Anderson and R.M. May, The invasion, persistence and spread of infectious diseases within animal and plant communities, Philos. Trans. R. Soc. Lond. B Biol. Sci., 314(1167) (1986), 533–570. https://doi.org/10.1098/rstb.1986.0072
R. Arditi and L.R. Ginzburg, Coupling in predator-prey dynamics: ratio-dependence, J. Theor. Biol., 139(3) (1989), 311–326. https://doi.org/10.1016/S0022-5193(89)80211-5
H. Baek, Species extinction and permanence of an impulsively controlled two prey one-predator system with seasonal effects, Biosystems, 98(1) (2009), 7–18. https://doi.org/10.1016/j.biosystems.2009.06.008
J.K. Baum and B. Worm, Cascading top‐down effects of changing oceanic predator abundances, Journal of Animal Ecology, 78(4) (2009), 699–714. https://doi.org/10.1111/j.1365-2656.2009.01531.x
S. Biswas, M. Saifuddin, S.K. Sasmal, S. Samanta, N. Pal, F. Ababneh and J. Chattopadhyay, A delayed prey-predator system with prey subject to the strong Allee effect and disease, Nonlinear Dynamics, 84 (2016), 1569–1594. https://doi.org/10.1007/s11071-015-2589-9
D.E. Burkepile and J.D. Parker, Recent advances in plant-herbivore interactions, F1000Research, 6 (2017), 119. https://f1000research.com/articles/6-119/v1
A. Casadevall and L. Pirofski, Host‐pathogen interactions: the attributes of virulence, The Journal of infectious diseases, 184(3) (2001), 337–344. https://doi.org/10.1086/322044
H.D. Crofton, A model of host-parasite relationships, Parasitology, 63(3) (1971), 343–364. https://doi.org/10.1017/S0031182000079890
P.H. Crowley and E.K. Martin, Functional responses and interference within and between year classes of a dragonfly population, J. North Am. Benthol. Soc., 8(3) (1989), 211–221. https://doi.org/10.2307/1467324
S. Gakkhar, B. Singh and R.K. Naji, Dynamical behavior of two predators competing over a single prey, Biosystems, 90(3) (2007), 808–817. https://doi.org/10.1016/j.biosystems.2007.04.003
D. Greenhalgh and M. Haque, A predator-prey model with disease in the prey species only, Math. Methods Appl. Sci., 30(8) (2007), 911–929. https://doi.org/10.1002/mma.815
K. Hadeler and H.I. Freedman, Predator-prey population with parasite infection, J. Math. Biol., 27(6) (1989), 609–631. https://doi.org/10.1007/BF00276947
J.K. Hale and S.M.V. Lunel, Introduction to functional differential equations, Vol. 99, Springer Science & Business Media, 2013. https://link.springer.com/book/10.1007/978-1-4612-4342-7
H.W. Hethcote, W. Wang, L. Han and Z. Ma, A predator-prey model with infected prey, Theor. Popul. Biol., 66(3) (2004), 259–268. https://doi.org/10.1016/j.tpb.2004.06.010
F.M. Hilker and K. Schmitz, Disease-induced stabilization of predator-prey oscillations, J. Theor. Biol., 255(3) (2008), 299–306. https://doi.org/10.1016/j.jtbi.2008.08.018
C.S. Holling, The functional response of predators to prey density and its role in mimicry and population regulation, Mem. Entomol. Soc. Can., 97(S45) (1965), 5–60. https://doi.org/10.4039/entm9745fv
R.D. Holt and J.H. Lawton, The ecological consequences of shared natural enemies, Annu. Rev. Ecol. Syst., 25 (1994), 495–520. https://www.jstor.org/stable/2097322
C. Ji and D. Jiang, Analysis of a predator-prey model with disease in the prey, Int. J. Biomath., 6(03) (2013), 1350012. https://doi.org/10.1142/S1793524513500125
Z. Kabata, Parasites and Diseases of Fish Cultured in the Tropics, Taylor and Francis, London, 1985. https://www.cabidigitallibrary.org/doi/full/10.5555/19850828300
T.K. Kar and S.K. Chattopadhyay, A focus on long-run sustainability of a harvested prey predator system in the presence of alternative prey, Comptes Rendus Biologies, 333(11-12) (2010), 841–849. https://doi.org/10.1016/j.crvi.2010.09.001
B.W. Kooi, G.A. van Voorn and K.P. Das, Stabilization and complex dynamics in a predator-prey model with predator suffering from an infectious disease, Ecol. Complex. 8(1) (2011), 113–122. https://doi.org/10.1016/j.ecocom.2010.11.002
A.J. Lotka, Elements of Physical Biology, Williams and Wilkins; Baltimore, New York, 1925. https://archive.org/details/elementsofphysic017171mbp/page/n25/mode/2up
C. MacNeil, J.T. Dick, M.J. Hatcher, et al., Parasite-mediated predation between native and invasive amphipods, Proc. R. Soc. Lond. Ser. B, Biol. Sci. 270(1521) (2003), 1309–1314. https://doi.org/10.1098/rspb.2003.2358
S. Mandal, A. Tripathi, R. Banerjee, F. Souna and P.K. Tiwari, Ecological and epidemiological ramification of fear: Exploring deterministic and stochastic dynamics in a predator-prey system with predator switching and harvesting, International Journal of Biomathematics, 2450109 (2024). https://doi.org/10.1142/S1793524524501092
Ö. Östman, J. Eklöf, B.K. Eriksson, J. Olsson, P.O. Moksnes, U. Bergström, Top‐down control as important as nutrient enrichment for eutrophication effects in North Atlantic coastal ecosystems, Journal of Applied Ecology, 53(4), (2016) 1138–1147. https://doi.org/10.1111/1365-2664.12654
T. Park, A matlab version of the Lyapunov exponent estimation algorithm of Wolf et al., Physica 16D, 1985. https://in.mathworks.com/matlabcentral/fileexchange/48084-wolf-lyapunov-exponent-estimation-from-a-time-series
R.O. Peterson and R.E. Page, Wolf density as a predictor of predation rate, Swed. Wildl. Res. (Sweden), 1, (1987) 771–773. https://agris.fao.org/search/en/providers/122495/records/647751c4f2e6fe92b364bfe9
S. Roy, S. Alam and J. Chattopadhyay, Role of nutrient bound of prey on the dynamics of predator-mediated competitive-coexistance, Biosystems, 82(2), (2005) 143–153. https://doi.org/10.1016/j.biosystems.2005.06.007
S. Roy, S. Samanta, H. Nayak and P.K. Tiwari, Deterministic vs. stochastic dynamics of a predator-prey system with disease in prey: Impact of fear and supplementary foods, Int. J. Biomath. 2450047 (2024). https://doi.org/10.1142/S1793524524500475
N. Sk, P.K. Tiwari, Y. Kang and S. Pal, A nonautonomous model for the interactive effects of fear, refuge and additional food in a prey-predator system, J. Biol. Syst., 29(01) (2021), 107–145. https://doi.org/10.1142/S0218339021500054
P.D. Spencer and J.S. Collie, A simple predator-prey model of exploited marine fish populations incorporating alternative prey, ICES J. Mar. Sci., 53(3) (1996), 615–628. https://doi.org/10.1006/jmsc.1996.0082
P.D.N. Srinivasu, B.S.R.V. Prasad and M. Venkatesulu, Biological control through provision of additional food to predators: A theoretical study, Theor. Popul. Biol., 72(1) (2007), 111–120. https://doi.org/10.1016/j.tpb.2007.03.011
P.D.N. Srinivasu and B.S.R.V. Prasad, Time optimal control of an additional food provided predator-prey system with applications to pest management and biological conservation, J. Math. Biol., 60, 591–613 (2010). https://doi.org/10.1007/s00285-009-0279-2
S.Y. Strauss, Indirect effects in community ecology: their definition, study and importance, Trends Ecol. Evol., 6(7) (1991), 206–210. https://doi.org/10.1016/0169-5347(91)90023-Q
P.C. van Rijn and M.W. Sabelis, Impact of plant-provided food on herbivore-carnivore dynamics, Plant-provided food for carnivorous insects: A protective mutualism and its applications, (2005) 223–266. https://doi.org/10.1017/CBO9780511542220.009
E. Venturino, Epidemics in predator-prey models: disease in the predators, Math. Med. Biol., 19(3) (2002), 185–205. https://doi.org/10.1093/imammb/19.3.185
V. Volterra, Variazioni e Fluttuazioni del Numero dIndivui in Specie Animali Conviventi, Memorie della R. Accademia Lincei, 2 (1926), 31–113. https://www.scribd.com/document/161177926/Variazioni-e-fluttuazioni-del-numero-d-individui-in-specie-animali-conviventi
A. Wolf, J.B. Swift, H.L. Swinney and J.A. Vastano, Determining Lyapunov exponents from a time series, Physica D: Nonlinear Phenomena, 16(3) (1985), 285–317. https://doi.org/10.1016/0167-2789(85)90011-9
Y. Zhang and J. Adams, Top‐down control of herbivores varies with ecosystem types, Journal of Ecology, 99(2) (2011), 370–372. https://doi.org/10.1111/j.1365-2745.2010.01770.x