Combined impact of fear and Allee effect in predator-prey interaction models on their growth

Al Amri, Kawkab Abdullah Nabhan and Khan, Qamar J A and Greenhalgh, David (2024) Combined impact of fear and Allee effect in predator-prey interaction models on their growth. Mathematical Biosciences and Engineering, 21 (10). pp. 7211-7252. ISSN 1551-0018 (https://doi.org/10.3934/mbe.2024319)

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Abstract

We develop and analyze two predator-prey models which incorporate llee and the fear effect on the prey growth rate. Firstly, we consider a predator-prey model where the predator predates the prey with a Holling type I functional response. We find four feasible equilibria of the model and examine their stabilities. By using the predator mortality rate as the bifurcation parameter, the model exhibits Hopf-bifurcation for the coexistence equilibrium. Furthermore, our numerical illustrations demonstrate the effect of fear and the Allee effect on the population densities and we found that after a particular time, the prey population has little impact because fear gradually grows as a result of habituation over time. The population of predators, however, declines as the fear intensity rises, indicating that the fear effect might result in a decline in the predator population. The dynamics of the delayed system are examined, and Hopf-bifurcation was discussed. Finally, we looked at an eco-epidemiological model that takes into account the cost of fear and the Allee effect. A predator-prey model in which the prey is afflicted with a disease is investigated. Susceptible prey and infected prey are the two types of prey in the overall population. Numerical simulations are carried out to show that as the Allee threshold rises, the uninfected prey and predator decrease, while the population of infected prey increases. When the Allee threshold hits a certain value, all populations become extinct. As fear intensity increases, the population of uninfected prey decreases, and beyond a certain level of fear, habituation prevents the uninfected prey from changing. After a certain level of fear, the system exhibits instability because the predator population goes extinct and as a result, there is only interaction between uninfected and infected prey, which increases disease transmission and so the infected prey increases. We discovered that under certain parametric conditions, the co-existence equilibrium undergoes a Hopf-bifurcation. We added a time delay to make the model more realistic, and Hopf-bifurcation is studied by taking the delay as the bifurcation parameter. When the delay passes through a series of critical values, Hopf-bifurcation is identied. We estimate the delay length to preserve stability.

ORCID iDs

Al Amri, Kawkab Abdullah Nabhan, Khan, Qamar J A and Greenhalgh, David ORCID logoORCID: https://orcid.org/0000-0001-5380-3307;
CORE (COnnecting REpositories)