Population dynamics with demographic variability is frequently studied using discrete random variables with continuous-time Markov chain (CTMC) models. An approximation of a CTMC model using continuous random variables can be derived in a straightforward manner by applying standard methods based on the reaction rates in the CTMC model. This leads to a system of Itô stochastic differential equations (SDEs) which generally have the form $$ d \mathbf {y} = \varvec{\mu } \, dt + G \, d \mathbf {W},$$ where $$\mathbf {y}$$ is the population vector of random variables, $$\varvec{\mu }$$ is the drift vector, and G is the diffusion matrix. In some problems, the derived SDE model may not have real-valued or nonnegative solutions for all time. For such problems, the SDE model may be declared infeasible. In this investigation, new systems of SDEs are derived with real-valued solutions and with nonnegative solutions. To derive real-valued SDE models, reaction rates are assumed to be nonnegative for all time with negative reaction rates assigned probability zero. This biologically realistic assumption leads to the derivation of real-valued SDE population models. However, small but negative values may still arise for a real-valued SDE model. This is due to the magnitudes of certain problem-dependent diffusion coefficients when population sizes are near zero. A slight modification of the diffusion coefficients when population sizes are near zero ensures that a real-valued SDE model has a nonnegative solution, yet maintains the integrity of the SDE model when sizes are not near zero. Several population dynamic problems are examined to illustrate the methodology.

Oncolytic virotherapy is an emerging treatment modality which uses replication-competent viruses to destroy cancers without causing harm to normal tissues. By the development of molecular biotechnology, many effective viruses are adapted or engineered to make them cancer-specific, such as measles, adenovirus, herpes simplex virus and M1 virus. A successful design of virus needs a full understanding about how viral and host parameters influence the tumor load. In this paper, we propose a mathematical model on the oncolytic virotherapy incorporating viral lytic cycle and virus-specific CTL response. Thresholds for viral treatment and virus-specific CTL response are obtained. Different protocols are given depending on the thresholds. Our results also support that immune suppressive drug can enhance the oncolytic effect of virus as reported in recent literature.

We present a proof of principle for the phenomenon of the tragedy of the commons that is at the center of many theories on the evolution of cooperation. Whereas the tragedy is commonly set in a game theoretical context, and attributed to an underlying Prisoner's Dilemma, we take an alternative approach based on basic mechanistic principles of species growth that does not rely on the specification of payoffs which may be difficult to determine in practice. We establish the tragedy in the context of a general chemostat model with two species, the cooperator and the cheater. Both species have the same growth rate function and yield constant, but the cooperator allocates a portion of the nutrient uptake towards the production of a public good -the "Commons" in the Tragedy- which is needed to digest the externally supplied nutrient. The cheater on the other hand does not produce this enzyme, and allocates all nutrient uptake towards its own growth. We prove that when the cheater is present initially, both the cooperator and the cheater will eventually go extinct, hereby confirming the occurrence of the tragedy. We also show that without the cheater, the cooperator can survive indefinitely, provided that at least a low level of public good or processed nutrient is available initially. Our results provide a predictive framework for the analysis of cooperator-cheater dynamics in a powerful model system of experimental evolution.

The host immune response can often efficiently suppress a virus infection, which may lead to selection for immune-resistant viral variants within the host. For example, during HIV infection, an array of CTL immune response populations recognize specific epitopes (viral proteins) presented on the surface of infected cells to effectively mediate their killing. However HIV can rapidly evolve resistance to CTL attack at different epitopes, inducing a dynamic network of interacting viral and immune response variants. We consider models for the network of virus and immune response populations, consisting of Lotka–Volterra-like systems of ordinary differential equations. Stability of feasible equilibria and corresponding uniform persistence of distinct variants are characterized via a Lyapunov function. We specialize the model to a “binary sequence” setting, where for n epitopes there can be 2^n distinct viral variants mapped on a hypercube graph. The dynamics in several cases are analyzed and sharp polychotomies are derived characterizing persistent variants. In particular, we prove that if the viral fitness costs for gaining resistance to each epitope are equal, then the system of 2^n virus strains converges to a “perfectly nested network” with less than or equal to n+1 persistent virus strains. Overall, our results suggest that immunodominance , i.e. relative strength of immune response to an epitope, is the most important factor determining the persistent network structure.

The article contains the author's reflections on recent developments in a very select portion of the now vast subject of monotone dynamical systems. Continuous time systems generated by cooperative systems of ordinary differential equations, delay differential equations, parabolic partial differential equations, and control systems are the main focus and results are included which the author feels have had a major impact in the applications. These include the theory of competition between two species or two teams and the theory of monotone control systems.

Marine ecosystems may contain diverse populations of bacteria which are supported by relatively few limiting nutrients and which are predated upon by an equally diverse set of bacteriophages (viruses). The network defined by which virus kill which bacteria strongly influences the overall persistence of the marine community. The dynamics of virus and immune response within a host can also be viewed as a complex ecological system. The immune response predates on the pathogen, and distinct viral strains compete for a target cell population, while immune response populations compete for the virus since their proliferation occurs upon pathogen recognition. For example, during HIV infection an extensive family of CTL immune cells recognize specific viral proteins presented on the surface of infected cells to effectively mediate their killing. However HIV can rapidly evolve resistance to CTL attack at different epitopes. The ensuing battle creates a dynamic network of interacting viral strains and immune response variants with variable levels of (strain) reactivity. It is unsurprising that similar mathematical models can be used to better understand the role of the predation network structure on the ultimate biodiversity of the community in each of these cases. The models we consider consist of systems of Lotka-Volterra-like ordinary differential equations. The talk will describe recent joint work with colleagues Dan Korytowski and Cameron Browne on the dynamics of such models.

We show that a bacteria and bacteriophage system with either a perfectly nested or a one-to-one infection network is permanent, a.k.a uniformly persistent, provided that bacteria that are superior competitors for nutrient devote the least to defence against infection and the virus that are the most efficient at infecting host have the smallest host range. By ensuring that the density-dependent reduction in bacterial growth rates are independent of bacterial strain, we are able to arrive at the permanence conclusion sought by Jover et al (J. Theor. Biol. 332:65-77, 2013). The same permanence results hold for the one-to-one infection network considered by Thingstad (Limnol Oceanogr 45:1320-1328, 2000) but without virus efficiency ordering. Additionally we show the global stability for the nested infection network, and the global dynamics for the one-to-one network.

We focus on the long-term dynamics of "killing the winner" Lotka-Volterra models of marine communities consisting of bacteria, virus, and zooplankton. Under suitable conditions, it is shown that there is a unique equilibrium with all populations present which is stable, the system is permanent, and the limiting behavior of its solutions is strongly constrained.

For a chemostat with time-dependent removal rates that may differ between species, it is shown that a microbial species dies out if there is another species that has both a lower break-even concentration and a “less concave” functional response. Conditions are also derived for microbial concentrations to remain bounded and the overall microbial population to persist.

In recent decades, marine ecologists have conducted extensive field work and experiments to understand the interactions between bacteria and bacteriophage (phage) in marine environments. This dissertation provides a detailed rigorous framework for gaining deeper insight into these interactions. Specific features of the dissertation include the design of a new deterministic Lotka-Volterra model with n + 1 bacteria, n/n + 1 phage, with explicit nutrient, where the j phage strain infects the first j bacterial strains, a perfectly nested infection network (NIN). This system is subject to trade-off conditions on the life-history traits of both bacteria and phage given in an earlier study Jover et al. (2013). Sufficient conditions are provided to show that a bacteria-phage community of arbitrary size with NIN can arise through the succession of permanent subcommunities, by the successive addition of one new population. Using uniform persistence theory, this entire community is shown to be permanent (uniformly persistent), meaning that all populations ultimately survive. It is shown that a modified version of the original NIN Lotka-Volterra model with implicit nutrient considered by Jover et al. (2013) is permanent. A new one-to-one infection network (OIN) is also considered where each bacterium is infected by only one phage, and that phage infects only that bacterium. This model does not use the trade-offs on phage infection range, and bacterium resistance to phage. The OIN model is shown to be permanent, and using Lyapunov function theory, coupled with LaSalle’s Invariance Principle, the unique coexistence equilibrium associated with the NIN is globally asymptotically stable provided that the interand intra-specific bacterial competition coefficients are equal across all bacteria. Finally, the OIN model is extended to a “Kill the Winner” (KtW) Lotka-Volterra model of marine communities consisting of bacteria, phage, and zooplankton. The zooplankton acts as a super bacteriophage, which infects all bacteria. This model is shown to be permanent. i

We show that the classical Volterra Lyapunov function can be used to obtain useful results for a Lotka–Volterra model of bacteria-virus infection networks. In particular, if a positive equilibrium exists, then it is stable and all positive trajectories are bounded and persistent in the sense that the limit inferior of each component is positive.

This volume is dedicated to the memory of Paul Waltman. Many of the authors of articles contained here were participantsat the NCTS International Conference on Nonlinear Dynamics with Applications to Biology heldMay 28-30, 2014 at National Tsing-Hua University, Hsinchu, Taiwan. The purpose of the conference was to survey new developmentsin nonlinear dynamics and its applications to biology and to honor the memory of Professor Paul Waltman for his influence on the development of Mathematical Biology and Dynamical Systems.Attendees at the conference included Paul's sons Fred and Dennis, many of Paul's former doctoral and post-doctoral students, many others who,although not students of Paul, nevertheless were recipients of Paul's valuable advice and council, and many colleagues from all over the worldwho were influenced by Paul's mathematics and by his personality. We thank the NCTS for its financial support of the conference and Dr. J.S.W. Wongfor supporting the conference banquet.For more information please click the “Full Text” above.

We study similarity solutions of a nonlinear partial differential equation that is a generalization of the heat equation. Substitution of the similarity ansatz reduces the partial differential equation to a nonlinear second-order ordinary differential equation on the half-line with Neumann boundary conditions at both boundaries. The existence and uniqueness of solutions is proven using Wazewski's Principle.

We provide sharp conditions distinguishing persistence and extinction for a class of discrete-time dynamical systems on the positive cone of an ordered Banach space generated by a map which is the sum of a positive linear contraction A and a nonlinear perturbation G that is compact and differentiable at zero in the direction of the cone. Such maps arise as year-to-year projections of population age, stage, or size-structure distributions in population biology where typically A has to do with survival and individual development and G captures the effects of reproduction. The threshold distinguishing persistence and extinction is the principal eigenvalue of \(({\mathbb {I}}-A)^{-1}G'(0)\) provided by the Krein-Rutman Theorem, and persistence is described in terms of associated eigenfunctionals. Our results significantly extend earlier persistence results of the last two authors which required more restrictive conditions on G. They are illustrated by application of the results to a plant model with a seed bank.

A model is considered for a spatially distributed population of male and female individuals that mate and reproduce only once in their life during a very short reproductive season. Between birth and mating, females and males move by diffusion on a bounded domain \(\Omega \) under Dirichlet boundary conditions. Mating and reproduction are described by a (positively) homogeneous function (of degree one). We identify a basic reproduction number \({\mathcal {R}}_0\) that acts as a threshold between extinction and persistence. If \({\mathcal {R}}_0 <1\), the population dies out while it persists (uniformly weakly) if \({\mathcal {R}}_0 > 1\). \({\mathcal {R}}_0\) is the cone spectral radius of a bounded homogeneous map.

We show that a chemostat community of bacteria and bacteriophage in which bacteria compete for a single nutrient and for which the bipartite infection network is perfectly nested is permanent, a.k.a. uniformly persistent, provided that bacteria that are superior competitors for nutrient devote the least effort to defence against infection and the virus that are the most efficient at infecting host have the smallest host range. This confirms an earlier work of Jover et al. (J. Theor. Biol. 332:65–77, 2013) who raised the issue of whether nested infection networks are permanent. In addition, we provide sufficient conditions that a bacteria-phage community of arbitrary size with nested infection network can arise through a succession of permanent subcommunties each with a nested infection network by the successive addition of one new population. The same permanence results hold for the monogamous infection network considered by Thingstad (Limnol Oceanogr 45:1320–1328, 2000) but without the trade-offs.

Biological systematics studies suggest that species are discretized in niche space. That is, rather than seeing a continuum of organism types with respect to continuous environmental variations, observers instead find discrete species or clumps of species, with one clump separated from another in niche space by a gap. Here, using a simple one dimensional model with a smoothly varying environmental condition, we investigate conditions for a discrete niche partitioning instability of a continuously varying species structure in the context of asexually reproducing microbes. We find that significant perturbation of translational invariance is required for instability, but that conditions for such perturbations might reasonably occur, for example, through influence of boundary conditions.

Oncolytic viruses are a form of cancer treatment used to target tumor cells without harming healthy cells. These viruses have been engineered to specifically infect and kill cancer cells. Maximizing oncolytic potential of replicating viruses, however, has not been found to be an optimal strategy, as opposed to maximizing viral spread through the tumor. The delay differential equation model in this work includes interactions of uninfected tumor cells, tumor cells infected by the virus, and virus specific antigens, representing an immune response. Here , the thresholds between replicating viruses parameters are explored to find the most optimal outcome towards the minimization of tumor cells.

This chapter, originally intended for inclusion in [4], focuses on modeling issues by way of an example of a predator-prey model where the predator has a juvenile stage. A careful derivation of the model is given starting from an age-structured model for the predator population. The method of characteristics is used to reduce the partial differential equations to a system of integro-differential equations which represent delay equations with unbounded delays. The modeling assumptions of fixed maturation period and exponentially distributed death rates reduces the model to a system of familiar discrete delay equations. However, the appropriate initial data for the resulting delay equation is not what is usually assumed in the literature. Rather than analyzing the system using analytical tools, we rely on numerical simulations to suggest that the new initial conditions are superior to those commonly employed. A mathematical analysis of the equations would be quite similar to that in Chapter 8 of [4].

We obtain sharp conditions distinguishing extinction from persistence and provide sufficient conditions for global stability of a positive fixed point for a class of discrete time dynamical systems on the positive cone of an ordered Banach space generated by a map which is, roughly speaking, a nonlinear, rank one perturbation of a linear contraction. Such maps were considered by Rebarber, Tenhumberg, and Towney (Theor. Pop. Biol. 81, 2012) as abstractions of a restricted class of density dependent integral population projection models modeling plant population dynamics. Significant improvements of their results are provided.