structured population dynamics
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Hosts diverge widely in how, and how well, they defend themselves against infection and immunopathology. Why are hosts so heterogeneous? Both epidemiology and life history are commonly hypothesized to influence host immune strategy, but the relationship between immune strategy and each factor has commonly been investigated in isolation. Here, we show that interactions between life history and epidemiology are crucial for determining optimal immune specificity and sensitivity. We propose a demographically-structured population dynamics model, in which we explore sensitivity and specificity of immune responses when epidemiological risks vary with age. We find that variation in life history traits associated with both reproduction and longevity alters optimal immune strategies–but the magnitude and sometimes even direction of these effects depends on how epidemiological risks vary across life. An especially compelling example that explains previously-puzzling empirical observations is that depending on whether infection risk declines or rises at reproductive maturity, later reproductive maturity can select for either greater or lower immune specificity, potentially illustrating why studies of lifespan and immune variation across taxa have been inconclusive. Thus, the sign of selection on the life history-immune specificity relationship can be reversed in different epidemiological contexts. Drawing on published life history data from a variety of chordate taxa, we generate testable predictions for this facet of the optimal immune strategy. Our results shed light on the causes of the heterogeneity found in immune defenses both within and among species and the ultimate variability of the relationship between life history and immune specificity.

Realistic models of biological processes typically involve interacting components on multiple scales, driven by changing environment and inherent stochasticity. Such models are often analytically and numerically intractable. We revisit a dynamic maximum entropy method that combines a static maximum entropy with a quasi-stationary approximation. This allows us to reduce stochastic non-equilibrium dynamics expressed by the Fokker-Planck equation to a simpler low-dimensional deterministic dynamics, without the need to track microscopic details. Although the method has been previously applied to a few (rather complicated) applications in population genetics, our main goal here is to explain and to better understand how the method works. We demonstrate the usefulness of the method for two widely studied stochastic problems, highlighting its accuracy in capturing important macroscopic quantities even in rapidly changing non-stationary conditions. For the Ornstein-Uhlenbeck process, the method recovers the exact dynamics whilst for a stochastic island model with migration from other habitats, the approximation retains high macroscopic accuracy under a wide range of scenarios in a dynamic environment.

This paper is devoted to study the null controllability properties of a population dynamics model with age structuring and nonlocal boundary conditions. More precisely, we consider a four-stage model with a second derivative with respect to the age variable. The null controllability is related to the extinction of eggs, larvae, and female population. Thus, we estimate a time T to bring eggs, larvae, and female subpopulation density to zero. Our method combines fixed point theorem and Carleman estimate. We end this work with numerical illustrations.

<p style='text-indent:20px;'>In this paper, we study the null controllability of a nonlinear age, space and two-sex structured population dynamics model. This model is such that the nonlinearity and the couplage are at birth level. We consider a population with males and females and we are dealing with two cases of null controllability problems.</p><p style='text-indent:20px;'>The first problem is related to the total extinction, which means that, we estimate a time <inline-formula><tex-math id="M1">\begin{document}$ T $\end{document}</tex-math></inline-formula> to bring the male and female subpopulation density to zero. The second case concerns null controllability of male or female subpopulation. Since the absence of males or females in the population stops births; so, if we have the total extinction of the females at time <inline-formula><tex-math id="M2">\begin{document}$ T, $\end{document}</tex-math></inline-formula> and if <inline-formula><tex-math id="M3">\begin{document}$ A $\end{document}</tex-math></inline-formula> is the life span of the individuals, at time <inline-formula><tex-math id="M4">\begin{document}$ T+A $\end{document}</tex-math></inline-formula> one will get certainly the total extinction of the population. Our method uses first an observability inequality related to the adjoint of an auxiliary system, a null controllability of the linear auxiliary system, and after the Schauder's fixed point theorem.</p>