International Journal of Biomathematics

In this paper, we introduce spatial memory into a predator–prey model with herd behavior. Taking memory-based diffusion coefficient and average memory period of predators as control parameters, we obtain the stable conditions of the positive equilibrium of system and prove the existence of Hopf bifurcation. In addition, a double Hopf bifurcation occurs at the intersection of the nonhomogeneous Hopf bifurcation curves, and a spatially nonhomogeneous quasi-periodic pattern can be observed near the double Hopf bifurcation point by numerical simulation.

The paper focuses on the study of an epidemic model for the evolution of diseases, using stochastic models. We demonstrated the encoding of this intricate model into formalisms suitable for analysis with advanced stochastic model checkers. A co-infection model’s dynamics were modeled as an Ito–Levy stochastic differential equations system, representing a compartmental system shaped by disease complexity. Initially, we established a deterministic system based on presumptions and disease-related traits. Through non-traditional analytical methods, two key asymptotic properties: eradication and continuation in the mean were demonstrated. Section 2 provides a detailed construction of the model. Section 3 results confirm that the outcome is biologically well-behaved. Utilizing simulations, we tested and confirmed the stability of all equilibrium points. The ergodic stationary distribution and extinction conditions of the system are thoroughly analyzed. Investigations were made into the stochastic system’s probability density function, and digital simulations were employed to illustrate the probability density function and systems’ extinction. Although infectious disease control and eradication are major public health goals, global eradication proves challenging. Local disease extinction is possible, but it may reoccur. Extinction is more feasible with a lower ${𝒦}$. Notably, our simulations showed that reducing the ${𝒦}$ value significantly increases the likelihood of disease extinction and reduces the probability of future recurrence. Additionally, our study provides insights into the conditions under which a disease can persist or become extinct, contributing to more effective disease control strategies in public health.

This paper has been motivated by the following ecological question: how influential is the water stress on plants production? The aim of this paper is to investigate the impact of water stress and irrigation on plants production. We propose a mathematical model for the dynamics growth of plants that takes into account the concentration of available water in the soil, the absorption of water by plants, water stress, the plants production and the plants compensation. We present the theoretical analysis of the model. More precisely, we show that the model is well-posed. We also compute trivial equilibria and derive a threshold parameter that determines the outcome of water stress within a plantation. Further, we perform a numerical simulation in the case of banana-plantain plant to support and valid theoretical results. We found that the Hopf bifurcation occurs for a specific value of the water absorption rate of unstressed plants. The impact of the water stress on the banana-plantain production is also numerically investigated. We found that the water stress can cause about $62.57\%$ tons of loss of banana-plantain production within a plantation with 1600 rejects initially planted. We also consider the irrigation as a control parameter of water stress. We found that the irrigation increases the stressed plant production to more than $60\%$ for $1600$ rejects initially planted. This suggests that climate change may play a detrimental role on banana-plantains production. We also investigate the role of the type of soil in banana-plantain production. Our finding suggests that the Fine Sandy Loam soil is more resilient to water stress than Silty Clay and Loamy soils.

This research aims to use fractional operators to analyze a fractional-order model of a coronavirus disease of 2019 (COVID-19). We use some basic results and definitions from fractional calculus and then, by using them, investigate the effects of these operators in a better elucidation of the epidemic COVID-19. We showed the existence and uniqueness of the solution of the proposed model by applying the Picard–Lindelöf theorem and Banach contraction principle. We established the generalized Hyers–Ulam stability of the fractional model using Gronwall’s inequality. We developed effective numerical scheme to solve these fractional systems, which gives a perfect approximate solution to the fractional system. Finally, the numerical simulations were performed in each case to verify all theoretical result.

Synergistic interaction between influenza and pneumonia is well established in the literature. In this study, we present a model for the transmission dynamics of co-infection with influenza and pneumococcal pneumonia, with the goal of assessing the effects of influenza co-infection on the transmission of pneumonia. We derive an expression for the basic reproductive number ${{ℛ}}_0=max({{ℛ}}_f,{{ℛ}}_p)$ where ${{ℛ}}_f$ and ${{ℛ}}_p$ are, respectively, the reproductive numbers for flu and pneumonia. We show that in the case ${{ℛ}}_f\le 1\le {{ℛ}}_p$, infection with influenza is driven to extinction while pneumonia is endemic, with the endemic state being globally asymptotically stable. The converse result holds in the case where ${{ℛ}}_p\le 1\le {{ℛ}}_f$. We also show the existence of the co-infection equilibrium. In this case, we show that the presence of co-infection results in a possible backward bifurcation in the system at ${{ℛ}}_0=1$; epidemiologically, this means that the spread of the infection will be harder to control. Numerical simulations are presented to verify the analytic results and gain further insights.

In this paper, we consider oncolytic M1 cancer virotherapy reaction–diffusion model for its numerical simulation. The Galerkin finite element method (FEM) based on cubic B-splines is setup to obtain numerical solutions of this nonlinear model. Stability of the corresponding linearized scheme is studied by the von Neumann method. Since the problem is nonlinear and its exact solutions are not available in the literature, obtained solutions are compared with the numerical solutions available in the literature. The proposed method is consistent and reliable to simulate oncolytic M1 cancer virotherapy reaction–diffusion model.

In this paper, a class of discrete-time predator–prey system with fear factor and harvesting effect is proposed, and its complex dynamic behavior is analyzed. First, the existence and stability of equilibrium points of the discrete system are studied. Second, the critical value expressions of Neimark–Sacker bifurcation and flip bifurcation are obtained by bifurcation theory. Third, we control the bifurcation and chaos of the system by using feedback control strategies. The optimal harvesting strategy is obtained by using the Pontryagin maximum principle. Finally, we use MATLAB software to carry out some numerical simulations, which not only verify our theoretical results, but also analyze the rich dynamic behavior of the system.

In this research, we examine a predator–prey model in which nonlocal fear plays a role alongside delay in a reaction–diffusion framework. We integrate two delays into the model to account for the lag between when fear starts affecting the growth rate of prey and when it starts affecting the growth rate of the predator through feedback. The first step is to investigate local and global stability and bifurcations in the equilibrium states of the nondelayed model. We explore the Hopf bifurcation in the delayed model using the delay as the bifurcation parameter. Our theoretical findings are then backed up by certain simulations. It reveals how the system, depending on its level of anxiety and the time delays involved, displays a wide range of spatiotemporal patterns.

A nonlinear dynamical model is developed in this paper that depicts interactions among vultures, human, animals and their carcasses. Diseases such as plague, anthrax and rabies are spreading due to the cascade impact caused by the catastrophic drop in the vulture population, particularly in Africa and Asia. The built model is theoretically studied using qualitative differential-equation theory to demonstrate the system’s rich dynamical features, which are critical for maintaining the ecosystem’s equilibrium. According to the qualitative findings, depending on the parameter combinations, the system not only displays stability of many equilibrium states but also experiences transcritical and Hopf bifurcations. By keeping an eye on the variables causing the decline of the vulture population, the model’s outputs may assist in maintaining balance with the prevention of the spread of disease through carcasses. Hopf-bifurcation results in the bifurcation of the limit cycle through the threshold, supporting the idea that interactions between humans and vultures may also be periodic.

In this paper, we study the stability of a constant equilibrium solution of the reaction–diffusion population equation under different boundary conditions through analysis of its characteristic equation. In a scalar reaction–diffusion equation, we have found that the stability of a constant equilibrium solution is different when the scalar reaction–diffusion equation is subject to Neumann boundary conditions, Dirichlet boundary conditions and the mixed type boundary conditions, respectively. Similarly, the more complex results are found in the two reaction–diffusion equations with all different kinds boundary conditions. The relevant numerical calculation results are carried out to demonstrate the validity of theoretical analysis.