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In this paper, a p53-Mdm2 mathematical model is analyzed to understand the biological implications of feedback loops in a p53 system. Results show that the model can undergo four types of codimension-3 Bogdanov–Takens bifurcations, including cusp, saddle, focus and elliptic. Specifically, we find new phenomena including the coexistence of four positive equilibria, two limit cycles, the coexistence of three stable states (two stable equilibria and one stable limit cycle, or three stable equilibria), a heteroclinic loop enclosing a smaller stable limit cycle and a larger stable limit cycle. These findings extend the understanding of the complex dynamics of the p53 system, and can provide some potential biological applications. [ABSTRACT FROM AUTHOR]

Scarlet fever is an acute respiratory infectious disease and the incidence rate is increasing from 2011 throughout the world. In this paper, the mathematical models are proposed, which incorporate both direct transmissions and indirect transmissions of scarlet fever. The threshold conditions for disease invasion are obtained in terms of the basic reproduction number. The peak value, final size and epidemic time in a seasonal prevalence are investigated numerically. Furthermore, the effects of seasonal fluctuations on disease outbreak are also studied on the basis of real data in China. [ABSTRACT FROM AUTHOR]

Immunotherapy using dendritic cells (DCs) as antigen-presenting cells is widely used in laboratory and animal studies as a promising treatment for advanced prostate cancer. In this paper, the bifurcation analysis of a nonlinear model of prostate cancer with immunotherapy is performed. It is found that the model exhibits complex behaviors such as a saddle-node bifurcation, Hopf bifurcation, Bogdanov–Takens bifurcation and generalized Hopf bifurcation. Moreover, it is shown that the vaccine amount and T-cells killing efficiency of tumor cells have a significant effect on tumor cells, which provides new insights to improve treatment outcomes even for patients with a weak immune system. [ABSTRACT FROM AUTHOR]

In this work, we consider a model with one basal resource and two species of predators feeding by the same resource. There are three non-trivial boundary equilibria. One is the saturated state E K of the prey without any predator. Other two equilibria, E1 and E2, are the coexistence states of the prey with only one species of predators. Using a high-dimensional shooting method, the Wazewski' principle, we establish the conditions for the existence of traveling wave solutions from E K to E2 and from E1 to E2. These results show that the advantageous species v2 always win in the competition and exclude species v1 eventually. Finally, some numerical simulations are presented, and biological interpretations are given. Copyright © 2016 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Saturated infection incidences and immune responses are incorporated into a mathematical model of malaria with two competitive strains of Plasmodium falciparum . The basic reproductive numbers of pathogens and the response numbers of host immunity are formulated. The complete classifications of global stability of the model are established in terms of these numbers by using the persistence theory and Lyapunov methods. It is found that two strains of parasites coexist within a host when the reproductive numbers and responsive numbers satisfy the explicit conditions defined by two inequalities, and undergo the competitive exclusion otherwise. [ABSTRACT FROM AUTHOR]

We study pattern formations in a predator-prey model with prey-taxis. It is proved that a branch of nonconstant solutions can bifurcate from the positive equilibrium only when the chemotactic is repulsive. Furthermore, we find the stable bifurcating solutions near the bifurcation point under suitable conditions. Copyright © 2014 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

The mathematical model is proposed to simulate the dynamics of rabies transmissions in the raccoon population where juveniles stay with their mother and become adults until they establish their own habitats. The basic reproduction number of rabies transmission is formulated and is shown to be a threshold value of disease invasion. The bifurcation direction from the disease-free equilibrium is proved to be forward when the basic reproduction number passes through unity for spatial homogenous environment. The global stability of the disease-free steady state is also studied. [ABSTRACT FROM AUTHOR]

Based on the new findings in a recent experimental study (Lee et al., Nature 467, 82-86, ) that antibiotic resistant mutants of bacteria produce indoles to protect the wild strain bacteria, we propose a mathematical model to describe the evolution of the wild strain, resistant strain and indoles with limited nutrient. We distinguish two cases: (i) mutation is negligible and a resistant strain preexists; (ii) mutation is not negligible. For (i), we establish conditions for co-persistence of both strains, which indicate that the wild strain can survive with the help from the altruistic resistant strain, whereas it dies out in the absence of such a benefit. This consolidates the experimental findings in Lee et al. (Nature 467:82-86 ). Further analysis and simulations also reveal some new phenomena not reported in Lee et al. (Nature 467:82-86 ), that is, periodic oscillations of the populations may occur within certain range of the parameters, and there exists bistability in the sense that a stable positive periodic solution coexists with a stable positive equilibrium. [ABSTRACT FROM AUTHOR]

The theory of the principal eigenvalue is developed for an elliptic eigenvalue problem associated with a linear parabolic cooperative system with some zero diffusion coefficients. Then the basic reproduction number and its computation formulae are established for reaction-diffusion epidemic models with compartmental structure. These theoretical results are applied to a spatial model of rabies to study the influence of spatial heterogeneity and population mobility on disease transmission. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we study a diffusive one-prey and two-predators system with Beddington–DeAngelis functional response. The sufficient and necessary conditions for the existence of coexistence states are obtained by means of the fixed point index theory. In addition, the stability and uniqueness of coexistence states are investigated. Finally, we give the sufficient conditions for extinction and permanence of the time-dependent system. [Copyright &y& Elsevier]

Some mathematical models are suggested to describe the tritrofic interactions among plants, herbivores and their carnivorous enemies attracted by defensive volatiles of plants. For the interactions of Volterra type, it is proved that the threshold value for the persistence of herbivore and carnivore populations is not affected by the chemical attractions. Furthermore, the attraction to carnivores is beneficial to reduce the density of herbivores and increase the density of plants. If the interaction of plants and herbivores takes the Leslie type, the model admits the fold bifurcation that induces bistable positive equilibria. Numerical computations indicate that the response time of carnivores to defensive volatiles of plants induces periodic cycles and irregular fluctuations. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we consider an epidemic model with the nonlinear incidence of a sigmoidal function. By mathematical analysis, it is shown that the model exhibits the bistability and undergoes the Hopf bifurcation and the Bogdanov–Takens bifurcation. By numerical simulations, it is found that the incidence rate can induce multiple limit cycles, and a little change of the parameter could lead to quite different bifurcation structures. [Copyright &y& Elsevier]

Abstract: Parental care is incorporated into a prey–predator model in which immature predators are taken care of by their parents. It is assumed that adult predators confront the problem to stay home to protect offspring or to go out to forage. The global dynamics of the mathematical model is analyzed by means of analytical methods and numerical simulations. Conditions for the extinction of predator populations are established and the manners in which predators become extinct are revealed. Bifurcation analysis shows that the model admits changes from the extinction of predators to stable coexistence at a positive equilibrium point, and then to stage-structure induced oscillations. It is shown that optimal invest of adult predators can be achieved. [Copyright &y& Elsevier]

Abstract: The model of three competitive populations with Gompertz growth is studied. The periodic solutions are ruled out by generalized Dulac criteria. On the basis of the analysis, we obtain conditions that ensure the asymptotic behavior of the model is simple. [Copyright &y& Elsevier]

An epidemic model is proposed to incorporate population dispersals between patches and to include a constant infection period. A basic reproduction number of the model is established by means of a next generation matrix. It is found that a disease may spread when the population migrates in two patches, even though it dies out in each isolated patch. It is also found that the disease admits multiple exchanges between persistence and extinction for some types of migrations of individuals. [ABSTRACT FROM AUTHOR]

Abstract: This paper studies population models with time delays that represent the dispersal time for individuals to move from one patch to other patches. Two kinds of threshold between the globally stable coexistence and extinction of the populations in all patches are obtained for an autonomous models and a periodic model. The autonomous case extends results of a model without any delays in paper [Y. Takeuchi, Acta. Appl. Math. 14 (1989) 49–57] to a model with dispersing time. It is shown that time delays decrease the threshold for the permanence of the populations in two ways. First, the populations go extinct for any length of time delays if they go extinct in the absence of the delays. Second, the time delays can drive the populations from coexistence to extinction even if the populations are permanent when there are no time delays. When environment in all patches is periodic in time, sufficient conditions for the existence and uniqueness of a globally stable positive periodic solution and sufficient conditions for the extinction of the populations are obtained. The results indicate that the “super”-rich food conditions can be removed. [Copyright &y& Elsevier]

Abstract: Mathematical models with stage structures are proposed to describe the process of awareness, evaluation and decision-making. First, a system of ordinary differential equations is presented that incorporates the awareness stage and the decision-making stage. If the adoption rate is bilinear and imitations are dominant, we find a threshold above which innovation diffusion is successful. Further, if the adoption rate has a higher nonlinearity, it is shown that there exist bistable equilibria and a region such that an innovation diffusion is successful inside and is unsuccessful outside. Secondly, a model with a time delay is proposed that includes an evaluation stage of a product. It is proved that the system exhibits stability switches. The bifurcation direction of equilibria is also discussed. [Copyright &y& Elsevier]

We propose a mathematical model to simulate the SARS outbreak in Beijing. The model consists of six subpopulations, namely susceptible, exposed, quarantined, suspect, probable and removed, as China started to report SARS cases as suspect and probable separately from April 27 and cases transferred from suspect class to probable class from May 2. By simplifying the model to a two-compartment suspect-probable model and a single-compartment probable model and using limited data, we are able to simulate the SARS outbreak in Beijing. We estimate that the reproduction number varies from 1.0698 to 3.2524 and obtain certain important epidemiological parameters. [Copyright &y& Elsevier]