Effects of mortality on a predator–prey model in crisp, fuzzy, and spatial environments: A dynamical approach

The dynamic relationship between predators and prey plays a vital role in upholding equilibrium within the natural environment. Mortality plays a crucial role in maintaining the delicate equilibrium of ecosystems. This paper delves into the consequences of mortality in a predator–prey model that incorporates hydra, the Allee effect, and mutual interference among predators. We first established a crisp predator–prey model and then transformed it into a fuzzy model, representing the control parameters as triangular intuitionistic fuzzy numbers. We transform the fuzzy model into the defuzzified model by applying a graded mean integration technique. This technique allows for efficient solution determination using triangular intuitionistic fuzzy numbers. The theoretical section explores the presence and durability of equilibrium points and Hopf bifurcation on mortality parameters. Living organisms have the ability to move from one place to another, so we created a spatial model based on a crisp model. In order to investigate the impact of random movement of species within a population in an isolated area with different mortality parameters, we employ Turing instability. We confirm the theoretical results using the MATLAB package. The phase trajectories for various initial conditions in both environments are displayed, showcasing the species’ population fluctuations. We use the MATCONT package to illustrate the various scenarios that emerge when we alter the mortality parameters. We calculate the presence of saddle–node (SN), Hopf point (H), and Cusp point (CS) in the model. In addition, our spatial model analysis reveals various spatial structures within the isolated domain, including spots, stripes, and mixed patterns. The results indicate that mortality has a beneficial impact on the prey–predator population, helping to sustain ecological balance.