Effect of Fear, Treatment, and Hunting Cooperation on an Eco-Epidemiological Model: Memory Effect in Terms of Fractional Derivative

Iran J Sci Technol Trans Sci https://doi.org/10.1007/s40995-022-01371-w (0123456789().,-volV)(0123456789().,-volV) RESEARCH PAPER Effect of Fear, Treatment, and Hunting Cooperation on an EcoEpidemiological Model: Memory Effect in Terms of Fractional Derivative Uttam Ghosh1 • Ashraf Adnan Thirthar2 • Bapin Mondal1 • Prahlad Majumdar1 Received: 14 May 2022 / Accepted: 30 September 2022 Ó The Author(s), under exclusive licence to Shiraz University 2022 Abstract In this paper, we have studied a fractional-order eco-epidemiological model incorporating fear, treatment, and hunting cooperation effects to explore the memory effect in the ecological system through Caputo-type fractional-order derivative. We have studied the behavior of different equilibrium points with the memory effect. The proposed system undergoes through Hopf bifurcation with respect to the memory parameter as the bifurcation parameter. We perform numerical simulations for different values of the memory parameter and some of model parameters. In the numerical results, it appears that the system is exhibiting a stable behavior from a period or chaotic nature with the increase in the memory effect. The system also exhibits two transcritical bifurcations with respect to the growth rate of the prey. At low values of prey’s growth, all species go to extinction, at moderate values of prey’s growth, only preys (susceptible and infected) can survive, and at higher values of prey’s growth, all species survive simultaneously. The paper ended with some recommendations. Keywords Eco-epidemic model  Fear effect  Hunting cooperation  Caputo fractional-order derivative  Transcritical bifurcation  Hopf bifurcation 1 Introduction In population dynamics, ecological interactions such as competition, mutualism, and predation, play an essential role. However, parasite infection also affects the size of populations. Thus, prey–predator interactions should not ignore this issue. There have been multiple field studies demonstrating parasitic infections in prey and predators. Parasites can reduce the ability of infected organisms to survive and reproduce by affecting their internal mechanisms. Therefore, we ought to be concerned about predator–prey systems in which both populations are infected. An eco-epidemiological approach focuses on infectious & Bapin Mondal 1 Department of Applied Mathematics, University of Calcutta, Kolkata 700009, India 2 Department of Studies and Planning, University of Fallujah, Anbar, Iraq diseases in populations and communities. The step-by-step process of analyzing a problem from a molecular, social, and demographic perspective is considered eco-epidemiology. A system’s dynamics are affected by infection in any part of the population or both populations. In recent years, infectious disease has emerged as a significant factor. Researchers are increasingly studying predator and prey with infectious diseases. In many studies, predator–prey models have been investigated only with a disease in the prey. Hethcote et al. (2004) presented a predator–prey model in which SIS parasitic infection in prey led to higher rates of predation on infected prey before predator predation. In Sinha et al. (2010), authors explore prey–predator interactions in the context of environmental toxicants and disease. According to their study, the toxicants affect the population, while the infected prey is much more vulnerable to the toxicants as well as predators than sound prey. Shaikh et al. (2021) investigates the dynamics of an eco-epidemic predator– prey system in which disease is spread to prey species and alternative food is provided to predators. Moustafa et al. 123 Iran J Sci Technol Trans Sci (2020) analyzes a fractional-order eco-epidemiological model with disease in the prey population and showed that the order of fractional derivative stabilizes the coexistence equilibrium. According to Meng et al. (2018), the predator population can survive in a predator–prey system with prey harvesting and disease spreading among prey species. Recently, Sk et al. (2022) have studied a prey–predator model that incorporated infection in prey in both deterministic and stochastic environments, and found that high levels of fear and low levels of refuge can eliminate the disease from the system. Some infectious diseases can influence the dynamics of predator–prey systems when they enter either the predator body or the prey body (Djilali and Ghanbari 2021) through some pathogens. Pathogens include germs, viruses, fungi, parasites, etc. The objects can spread by direct or indirect contact with animals or some other way such as water and air (Van Seventer and Hochberg 2017). If the infection is not treated, it may be harmful. According to the Food and Agriculture Organization of the United Nations (FAO) (Romain et al. 2020), livestock accounts for ð40%Þ of the total agricultural production in developed countries, and for ð20%Þ in developing countries. The infection agencies may directly infect the prey or the predator, or the predator may become infected after consumption of the infected prey (Andrew et al. 2016). An important topic in ecological systems with epidemics is the dynamic relationship between predators and their prey. Eco-epidemiological models are investigated the ecological system with infection (Mukherjee 2010; Chakraborty et al. 2011; Chattopadhyay et al. 2002, 1999). The necessity of conserving wild animals has led many ecologists and eco-epidemiologists to become familiar with ecoepidemiology. Eco-epidemiological models discuss the prey–predator relationships when some of the species are infected (Juneja and Agnihotri 2018). One of the main objectives of the investigation of the eco-epidemic model is to control the spreading of diseases when disease and treatment both coexist simultaneously. Many studies have been published on ecological and epidemiological models with the disease either in prey (Meng et al. 2018; Mortoja et al. 2018) or in predator (Rana et al. 2016; Juneja and Agnihotri 2018) or in both prey and predator (Agnihotri and Juneja 2015; Hsieh and Hsiao 2008) and reference therein. Several field survey data and experimental results on terrestrial vertebrates showed that the fear of predators would cause a substantial variability of prey density (Sarkar and Khajanchi 2020). In Mukherjee (2020); Zhang et al. (2019), the authors have studied some of these types of models in the presence of fear effect and competitor for the prey in the predator–prey model with prey refuge. Treatment of the infected prey populations restores the 123 prey to its previous situation; as a result, availability of susceptible prey becomes plenty to the predator, and dynamics of the system may be more complex compared to other situations (Adnan Thirthar 2020). There are ample information in the existing literature on predator–prey interactions, which utilize the diversity of functional responses of both prey and (...truncated)