Narrow Results

In this paper, we present and assess a predator–prey Leslie–Gower model including disease, refuge and treatment in prey population. There are two groups of prey: those who are susceptible and infected. It is hypothesized that prey population is affected by diseases and refuge, and grows logistically in the absence of predators. Infected prey population receives treatment. The predators’ growth rate is governed by the modified Leslie–Gower dynamics. The dynamical attributes of the resulting system are boundedness, positivity of solutions, extinction criteria, existence and (local and global) stability. Biology uses mathematical analysis to identify the possible attributes of equilibrium points. The focus of this study is to assess how treatment and refuge affect the populations of ill prey, susceptible prey, predators and treated prey. The numerical simulation indicates that the influence of treatment, and refuge change the dynamics of the system (2.1). Extensive numerical simulations were performed to validate our analytical findings by using the Mathematica and MATLAB software.

Inability to become pregnant after 12 months of regular, unprotected intercourse is defined as Infertility. Couples who have not conceived after 12 months of unprotected vaginal intercourse should be offered further evaluation. Evaluation includes workup for anovulation (hormonal evaluations), hysterosalpingography, hysteroscopy, Laparoscopy. Treatment options vary from low-cost non-pharmacological therapy, like counseling on the timing of intercourse during the most fertile period, may wait for another year of unprotected intercourse, weight loss (target Body Mass Index $<30$), smoking cessation, limiting alcohol consumption, psychological interventions like cognitive behavioral therapy (CBT) to reduce stress, anxiety and depression related to infertility, to high-cost pharmacological approach including ovulation induction medication, intrauterine insemination (IUI), in vitro fertilization (IVF). Mathematical models are rising as a key factor to add to our knowledge of the fertility process and help us understand the intricacies in the reproductive system to be able to predict the possibilities of pregnancy precisely. We have created a mathematical model with five compartments to understand the success of treatment of infertility in women. We have carried out local stability, global stability at pregnancy-free and pregnancy exist equilibrium points and numerical analysis. We have also tried optimal control by maximizing fertility through non-pharmacological measures and applied cost control to IVF treatment. Our results showed non-pharmacological and pharmacological treatments have a positive impact on the overall success of treatment of infertility however cost is the important determining factor. We recommend maximizing non-pharmacological measures before opting for costly pharmacological measures. We also recommend that the government or other Non-Governmental Organizations (NGOs) help with the cost for women with infertility.

The bouncing ball system with two rigidly connected harmonic limiters is revisited in order to further analyze its periodic movement and bifurcation dynamics. By using the impulsive impact maps, we obtain several sufficient conditions for the existence and local stability of three different types of periodic orbits, respectively, and then plot the bifurcation diagrams in the space of the relative velocity and the restitution coefficient for different parameters of the limiter. The numerical simulation results are consistent with those of the theoretical analysis.

In this paper, we discuss in detail the local stability, Hopf bifurcation and pitchfork bifurcation of an inertial two-neuron system with time delays by applying the second-order approximation approach and analyzing the associated transcendental equation, respectively. Comparison has been made between the two methods. Numerical results have been presented to verify the analytical predictions. Specially, numerical examples show the time-varying control and the impulsive control which can both improve the stability of the system effectively.

This paper describes a cholera disease transmission model in the human population through the consumption of zooplankton as food by humans. Here the plankton population is classified into two subpopulations such as phytoplankton and zooplankton. Also, human population is classified into two subpopulations such as susceptible human and infected human. The proposed system reflects the impacts of using time delay in the cholera disease transmission. Different possible equilibrium points of our proposed system have been determined. Here local and global stabilities of our proposed system have been analyzed. The existence of Hopf bifurcation has been studied at the interior equilibrium point. The normal form method and center manifold theorem have been used to test the nature of Hopf bifurcation. It is observed that the interior equilibrium is locally asymptotically stable when the time delay in disease transmission term is large, while the change of stability of positive equilibrium will cause a bifurcating periodic solution at the time delay $\tau$ to be at less than its critical value. Finally, some numerical simulation results have been presented for the better understanding of our proposed system.

Chaotic phenomena may exist in nonlinear evolution equations. In many cases, they are undesirable but can be controlled. In this study, we deal with the chaos control of a three-dimensional chaotic system, reduced from a KdV–Burgers–Kuramoto equation. By adding a single delay feedback term into the chaotic system, we investigate the local stability and occurrence of Hopf bifurcation near the equilibrium point. Some dynamical properties including the direction and stability of bifurcated periodic solutions are presented by using the normal form theory and the center manifold theorem. Numerical simulations are illustrated which agree well with the theoretical results.

Social insect colonies’ robust and efficient collective behaviors without any central control contribute greatly to their ecological success. Colony migration is a leading subject for studying collective decision-making in migration. In this paper, a general colony migration model with Hill functions in recruitment is proposed to investigate the underlying decision making mechanism and the related dynamical behaviors. Our analysis provides the existence and stability of equilibrium, and the global dynamical behavior of the system. To understand how piecewise functions and Hill functions in recruitment impact colony migration dynamics, the comparisons are performed in both analytic results and bifurcation analysis. Our theoretical results show that the dynamics of the migration system with Hill functions in recruitment differs from that of the migration system with piecewise functions in the following three aspects: (1) all population components in our colony migration model with Hill functions in recruitment are persistent; (2) the colony migration model with Hill functions in recruitment has saddle and saddle-node bifurcations, while the migration system with piecewise functions does not; (3) the system with Hill functions has only equilibrium dynamics, i.e. either has a global stability at one interior equilibrium or has bistablity among two locally stable interior equilibria. Bifurcation analysis shows that the geometrical shape of the Hill functions greatly impacts the dynamics: (1) the system with flatter Hill functions is less likely to exhibit bistability; (2) the system with steeper functions is prone to exhibit bistability, and the steady state of total active workers is closer to that of active workers in the system with piecewise function.

The rarity of species increases its market price, consequently leading to the overexploitation of the species and even the extinction of the species. We study how the harvest intensity and the additive Allee effect impact on the Gordon–Schaefer model. In addition, by Sotomayor’s theorem and Poincaré–Andronov theorem, we prove the existence of Hopf bifurcation, saddle-node bifurcation and transcritical bifurcation, respectively. Finally, we illustrate our results by numerical simulations. We find that both the cost per unit of harvest and the additive Allee effect have a significant impact on human exploitation of the population. As the additive Allee effect reduces to the weak Allee effect, the lower harvest cost encourages humans to increase the exploitation of species. This threshold is a switch that controls the strong Allee effect. If it exceeds its threshold, then the motivation of humans to exploit the species increases.

The role of viruses in marine phytoplankton-zooplankton community structure is undoubtedly very important. In this paper, we propose a simple mathematical model for phytoplankton-zooplankton (prey-predator) system with an additional factor that the viral disease is spreading only among the prey species. Considering high abundance and importance of viruses in aquatic environments we have explicitly considered here the growth equation of free viruses and have studied this four-dimensional model analytically. It is observed that the disease-free system can be obtained when the virus replication factor lies in-between certain critical values. Numerical simulations have also been performed to substantiate the analytical findings.

The Leslie–Gower predator–prey model with logistic growth in prey is here modified to include an SI parasitic infection affecting the prey population only. Thresholds are identified for the predator population to survive, and the conditions for the disease to die out naturally are given. The behavior of the system around each equilibrium is investigated, showing that the disease incidence may have a relevant influence on the dynamics of complex ecosytems, assuming at times the role of a biological control parameter.

In this paper we consider a modification of the model of angiogenesis process proposed by Agur et al., in Discrete & Cont. Dyn. Sys. B4(1): 29–38, 2004. The number of steady states and their stability depending on the model parameters are studied. The hysteresis effect and cusp catastrophe are found for some parameters. The effect of hysteresis when the number of positive steady states changes from one to three is studied. The time delays are introduced. Numerical simulations, which show that oscillatory behaviour is possible, are performed.

The transmission of Plasmodium falciparum and Plasmodium vivax malaria in a mixed population of Thais and migrant Burmese living along the Thai-Myanmar border is studied through a mathematical model. The population is separated into two groups: Thai and Burmese. Each population in turn is divided into susceptible, infected, recovered and in case of vivax infection, a dormant subclass. The model is then modified to allow for some of the Burmese (given as a fraction P) to be infectious when they enter into Thailand. The behaviour of the modified model is obtained using a standard dynamical analysis. A new basic reproduction number is obtained. Numerical simulations of the modified model show that when P ≠ 0 and the same set of parameter values used in the initial model are used, the Thai population will be in the epidemic state. In other words, the repeated introduction of infectious Burmese (no matter how small of a number) will result in a malaria epidemic among the Thais irregardless of the public health practice undertaken by the Thai government. In the presence of the infected Burmese, the Thai government would have to increase the facilitites to treat the people who are infected by the malaria.

In this paper, a three-species food chain model has been developed by considering the interaction between prey, predator and super-predator species. It is assumed that in the absence of predator and super-predator species, the prey species grow logistically. It is also assumed that predator and super-predator consume prey and predator, respectively. It is assumed that the predator shows refuge behavior to the super-predator. Again, harvesting of super-predator population has been considered. It is assumed that the consumption of prey and predator follows Crowley–Martin-type functional form. Boundedness of the solution of the system has been studied and different equilibrium points are determined and the stability of the system around these equilibrium points has been investigated. Existence conditions of Hopf bifurcation with respect to ${\gamma }_1$ of the system have been studied. It is found that the system shows some complex and critical dynamics due to increase of handling time of prey. It is also found that the system moves towards stable steady state due to increase of predator interference. It is observed that predator refuge may be responsible for the stability of the system. The chaotic dynamics of the system have been found due to the increase of the harvesting rate of super-predator.

In this paper, we proposed and analyzed a nonlinear deterministic model for the impact of temperature variability on the epidemics of the malaria. The model analysis showed that all solutions of the systems are positive and bounded with initial conditions in a certain set. Thus, the model is proved to be both epidemiologically meaningful and mathematically well-posed. Using the next-generation matrix approach, the basic reproduction number with respect to the disease-free equilibrium (DFE) point is obtained. The local stability of the equilibria points is shown using the Routh–Hurwitz criterion. The global stability of the equilibria points is performed using the Lyapunov function. Also, we proved that if the basic reproduction number is less than one, the DFE is locally and globally asymptotically stable. But, if the basic reproduction number is greater than one, the unique endemic equilibrium exists, locally and globally asymptotically stable. The sensitivity analysis of the parameters is also described. Moreover, we used the method implemented by the center manifold theorem to identify that the model exhibits forward and backward bifurcations. From our analytical results, we confirmed that the variation of temperature plays a significant role on the transmission of malaria. Lastly, numerical simulations are demonstrated to enhance the analytical results of the model.

Considering that rumors propagation is affected by many factors in real life, based on the SIRS infectious disease model in complex networks, an extended ISRI rumor propagation model is proposed by using the probability function to define the influence mechanisms such as trust mechanism, and suspicion mechanism. First, dynamic equations are established for homogeneous and heterogeneous networks, and the rumor and rumor-free equilibrium points in the two networks are analyzed, respectively. Then, the basic reproduction number $R_0$ is obtained by using the next generation matrix and derivative calculation methods. Next, the lyapunov function is constructed to discuss the local stability and global stability of the equilibrium point, and the influence of different parameters on the basic reproduction number $R_0$. In addition, we selected ER network and BA network and found that population flow has a significant impact on the speed and scale of rumor propagation. At the same time, the trust mechanism can improve the propagation speed and scale, while the skepticism mechanism can inhibit the propagation speed, and it is more obvious in the BA network. The interaction between these mechanisms further affects the propagation characteristics of rumors in the network.

In this paper, a cholera disease transmission mathematical model has been developed. According to the transmission mechanism of cholera disease, total human population has been classified into four subpopulations such as (i) Susceptible human, (ii) Exposed human, (iii) Infected human and (iv) Recovered human. Also, the total bacterial population has been classified into two subpopulations such as (i) Vibrio Cholerae that grows in the infected human intestine and (ii) Vibrio Cholerae in the environment. It is assumed that the cholera disease can be transmitted in a human population through the consumption of contaminated food and water by Vibrio Cholerae bacterium present in the environment. Also, it is assumed that Vibrio Cholerae bacterium is spread in the environment through the vomiting and feces of infected humans. Positivity and boundedness of solutions of our proposed system have been investigated. Equilibrium points and the basic reproduction number $(R_0)$ are evaluated. Local stability conditions of disease-free and endemic equilibrium points have been discussed. A sensitivity analysis has been carried out on the basic reproduction number $(R_0)$. To eradicate cholera disease from the human population, an optimal control problem has been formulated and solved with the help of Pontryagin’s maximum principle. Here treatment, vaccination and awareness programs have been considered as control parameters to reduce the number of infected humans from cholera disease. Finally, the optimal control and the cost-effectiveness analysis of our proposed model have been performed numerically.

In this paper, the cooperating two-species Lotka–Volterra model is discussed. The authors study the existence of solutions to a strongly coupled elliptic system with homogeneous Dirichlet boundary conditions and consider the existence, stability, and global attractivity of time-periodic solutions for a coupled parabolic equations in a bounded domain. Their results show that this model possesses at least one coexistence state if cross-diffusions are weak. The existence of the positive T-periodic solutions, the local stability, and the global attractivity for the parabolic system are also given.

The dynamical behaviors of a two-species discrete ratio-dependent predator–prey system are considered. Some sufficient conditions for the local stability of the equilibria is obtained by using the linearization method. Further, we also obtain a new sufficient condition to ensure that the positive equilibrium is globally asymptotically stable by using an iteration scheme and the comparison principle of difference equations, which generalizes what paper [G. Chen, Z. Teng and Z. Hu, Analysis of stability for a discrete ratio-dependent predator–prey system, Indian J. Pure Appl. Math.42(1) (2011) 1–26] has done. The method given in this paper is new and very resultful comparing with papers [H. F. Huo and W. T. Li, Existence and global stability of periodic solutions of a discrete predator–prey system with delays, Appl. Math. Comput.153 (2004) 337–351; X. Liao, S. Zhou and Y. Chen, On permanence and global stability in a general Gilpin–Ayala competition predator–prey discrete system, Appl. Math. Comput.190 (2007) 500–509] and it can also be applied to study the global asymptotic stability for general multiple species discrete population systems. At the end of this paper, we present an open question.

Based on an epidemic model which Manvendra and Vinay [Mathematical model to simulate infections disease, VSRD-TNTJ3(2) (2012) 60–68] have proposed, we consider the dynamics and therapeutic strategy of a SEIS epidemic model with latent patients and active patients. First, the basic reproduction number is established by applying the method of the next generation matrix. By means of appropriate Lyapunov functions, it is proven that while the basic reproduction number 0 < R₀ < 1, the disease-free equilibrium is globally asymptotically stable and the disease eliminates; and if the basic reproduction number R₀ > 1, the endemic equilibrium is globally asymptotically stable and therefore the disease becomes endemic. Numerical investigations of their basin of attraction indicate that the locally stable equilibria are global attractors. Second, we consider the impact of treatment on epidemic disease and analytically determine the most effective therapeutic strategy. We conclude that the most effective therapeutic strategy consists of treating both the exposed and the infectious, while treating only the exposed is the least effective therapeutic strategy. Finally, numerical simulations are given to illustrate the effectiveness of the proposed results.

We consider a model of the exploitative competition of two micro-organisms for two complementary nutrients in a chemostat and take into account the interspecific interaction. The growth functions occurring in the model are of general type and the interaction functions are monotonic and positive. By the mean of the Thieme–Zhao theorem, we establish conditions for uniform persistence of the model.