Pinar Baydemir

We consider a general Leslie type prey-predator system involving harvesting and refuge effects. In the model, we use a general function to define how the population of prey grows. In this paper, we explore dynamical behavior of this system by taking three different growth forms for prey, namely, (a) exponential growth, (b) logistic growth and (c) logistic growth including Allee effect, respectively. For each model, we present global stability analysis by applying linear stability theory, utilizing Bendixson-Dulac criteria and constructing an appropriate Lyapunov function, respectively. From a mathematical point of view, we have concluded that harvesting and refuge have no effects on global stability of each model's consistent condition. Furthermore, we investigate the impacts of an external harvesting and a refuge for prey population on densities of prey and predator species. Mathematical analysis yields the result that prey population density decreases under the harvesting effect and increases under the refuge effect, while predator population density decreases as the harvesting and refuge effects increase. At last, in order to support analytical results and also to explore further impacts of prey refuge and harvesting on dynamics, we perform numerical simulations. We numerically discuss their impacts on the densities in early time. The results obtained underline that these effects are important for the continuation of the balanced functioning between prey and predator species, so they should be accounted into prey-predator models together.

The Allee effect is an important biological concept with a long history and broad interdisciplinary reach into mathematics and related fields. In this brief Perspective, we examine the pronounced mismatch between the limited evidence for Allee effects in the real world and the intense attention they receive in the theoretical literature. Our examination suggests that this theoretical enthusiasm is often driven less by biological facts than by the mathematical appeal of the dynamical complexity that Allee-type mechanisms introduce into models. The challenges inherent in empirical studies—including limited observation windows and the difficulty of isolating low-density effects within transient dynamics—have constrained empiricists’ attention to Allee effects in connection with spatial processes, whereas this same interface is well studied in the theoretical literature. To help bridge this gap between theoretical studies and field observations, we advocate greater use of stochastic spatial models that can link processes such as local extinction, spatial spread, and pattern formation to the kinds of biological data available from real systems. Tighter connections are needed between theoretical models and the biological processes that inspire them, and mathematical research identifying spatial signatures of Allee effects would help clarify the importance of those effects in nature.

We investigate a Leslie-type prey–predator system with an Allee effect to understand the dynamics of populations under stress. First, we determine stability conditions and conduct a Hopf bifurcation analysis using the Allee constant as a bifurcation parameter. At low densities, we observe that a weak Allee effect induces a supercritical Hopf bifurcation, while a strong effect leads to a subcritical one. Notably, a stability switch occurs, and the system exhibits multiple Hopf bifurcations as the Allee effect varies. Subsequently, we perform a sensitivity analysis to assess the robustness of the model to parameter variations. Additionally, together with the numerical examples, the FAST (Fourier amplitude sensitivity test) approach is employed to examine the sensitivity of the prey–predator system to all parameter values. This approach identifies the most influential factors among the input parameters on the output variable and evaluates the impact of single-parameter changes on the dynamics of the system. The combination of detailed bifurcation and sensitivity analysis bridges the gap between theoretical ecology and practical applications. Furthermore, the results underscore the importance of the Allee effect in maintaining the delicate balance between prey and predator populations and emphasize the necessity of considering complex ecological interactions to accurately model and understand these systems.