Functional responses and interference within and between year classes of a dragonfly population. Some Characteristics of Simple Types of Predation and Parasitism. The natural control of animal populations. Introduction to Population Ecology Cambridge University Press: Cambridge, UK, 2006. Basic properties of mathematical population models. Variazione e fluttuazini del numero d’individui in specie animali conviventi. Elements of Physical Biology Williams & Wilkins: Baltimore, MD, USA, 1925. Prey–Predator Models with Variable Carrying Capacity. Dynamics of a two predator-one prey system. A ratio-dependent food chain model and its applications to biological control. Mathematical Biology Springer: New York, NY, USA, 2002 Volume 2. Global qualitative analysis of a ratio-dependent predator-prey system. Predator-prey population with parasite infection. The author declares no conflict of interest. ![]() Moreover, in the last case (i.e., i = 3.0), when neglecting the oscillation parameter i.e., ϵ = 0, the likelihood of the coexistence of the predator–prey system increases more because the joint equilibrium point of the predator and prey increases and moves away from the axes where the predator and prey isoclines cross as displayed through Figure 5b, in addition, the densities of the prey and predator increase as shown in Figure 5a. It was interpreted that the likelihood of the coexistence of the predator–prey system increases as the value of i increases. ![]() On the other side, when the value of the immigration parameter ( i) increases, the joint equilibrium point of predator and prey increases and moves away from the axes where the predator and prey isoclines cross as shown through Figure 2b, Figure 3b, Figure 4b and Figure 5b. However, when investigating the oscillation of immigration of the prey, the dynamic behaviors of the model (32) tend to exhibit stable fluctuated and the fluctuations increase because of the increasing immigration parameter and the oscillation parameter that exists within the immigration, which is displayed through Figure 2, Figure 3 and Figure 4. There is an important question that can be asked in this context: how does adding the oscillation of immigration of the prey affect the dynamic behaviors? Figure 1, Figure 2, Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9 and Figure 10 illustrate the different kinds of graphs that are used to explain these cases time series figures and zero-growth isoclines with phase plane trajectory figures are plotted to show the dynamic behaviors and trajectories comprehensively, as well as the cross-sectional picture of zero-growth isoclines with phase plane trajectories for each figure is introduced to give clear picture of the dynamic behavior. In Section 3, the theoretical analysis presents two different dynamics of the model (1), so two different sets of the hypothetical values of the parameters were selected for representing two different dynamic behaviors that were steady state and fluctuated, respectively. They were assumed to satisfy the theoretical side of each case, but the values of the initial conditions were fixed for all cases. “NDsolve” command in the MATHEMATICA 11.3 software package was used to solve the models numerically for different sets of the hypothetical values of the parameters. The numerical simulations of models (1) and (32) were performed to show the change in dynamic behaviors and explain the effects that come from investigating the oscillation of the immigration of the prey. In addition, the Crowley–Martin type has been investigated in stochastic predator–prey models for studying the asymptotic properties of these models. Moreover, the Crowley–Martin type was used with a discrete predator–prey system to study the complex dynamics that lead to chaos. used the Crowley–Martin type in a delayed predator–prey model involving disease in the prey population. Alebraheem and Abu-Hassan investigated the seasonality in the predator–prey model with the Crowley–Martin type to identify complex dynamic behaviors. ![]() Stage-structured predator–prey models with the Crowley–Martin type have been widely considered to discuss the dynamics of these models. Ali and Jazar studied the global dynamics of a modified Leslie–Gower predator–prey model using the Crowley–Martin functional response and Holling type II as the numerical response. Upadhyay and Naji used Holling type II and the Crowley–Martin type with a three-species food chain model to discuss the dynamics of the model. In the literature, few studies have investigated the Crowley–Martin type of responses in predator–prey models using different concepts to study the dynamics of these models.
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