Journal: Journal of mathematical biology
We study the dynamics of a predator-prey system where predators fight for captured prey besides searching for and handling (and digestion) of the prey. Fighting for prey is modelled by a continuous time hawk-dove game dynamics where the gain depends on the amount of disputed prey while the costs for fighting is constant per fighting event. The strategy of the predator-population is quantified by a trait being the proportion of the number of predator-individuals playing hawk tactics. The dynamics of the trait is described by two models of adaptation: the replicator dynamics (RD) and the adaptive dynamics (AD). In the RD-approach a variant individual with an adapted trait value changes the population’s strategy, and consequently its trait value, only when its payoff is larger than the population average. In the AD-approach successful replacement of the resident population after invasion of a rare variant population with an adapted trait value is a step in a sequence changing the population’s strategy, and hence its trait value. The main aim is to compare the consequences of the two adaptation models. In an equilibrium predator-prey system this will lead to convergence to a neutral singular strategy, while in the oscillatory system to a continuous singular strategy where in this endpoint the resident population is not invasible by any variant population. In equilibrium (low prey carrying capacity) RD and AD-approach give the same results, however not always in a periodically oscillating system (high prey carrying-capacity) where the trait is density-dependent. For low costs the predator population is monomorphic (only hawks) while for high costs dimorphic (hawks and doves). These results illustrate that intra-specific trait dynamics matters in predator-prey dynamics.
During an infection, HIV experiences strong selection by immune system T cells. Recent experimental work has shown that MHC escape mutations form an important pathway for HIV to avoid such selection. In this paper, we study a model of MHC escape mutation. The model is a predator-prey model with two prey, composed of two HIV variants, and one predator, the immune system CD8 cells. We assume that one HIV variant is visible to CD8 cells and one is not. The model takes the form of a system of stochastic differential equations. Motivated by well-known results concerning the short life-cycle of HIV intrahost, we assume that HIV population dynamics occur on a faster time scale then CD8 population dynamics. This separation of time scales allows us to analyze our model using an asymptotic approach. Using this model we study the impact of an MHC escape mutation on the population dynamics and genetic evolution of the intrahost HIV population. From the perspective of population dynamics, we show that the competition between the visible and invisible HIV variants can reach steady states in which either a single variant exists or in which coexistence occurs depending on the parameter regime. We show that in some parameter regimes the end state of the system is stochastic. From a genetics perspective, we study the impact of the population dynamics on the lineages of an HIV sample taken after an escape mutation occurs. We show that the lineages go through severe bottlenecks and that in certain parameter regimes the lineage distribution can be characterized by a Kingman coalescent. Our results depend on methods from diffusion theory and coalescent theory.
Many ecological systems exhibit multi-year cycles. In such systems, invasions have a complicated spatiotemporal structure. In particular, it is common for unstable steady states to exist as long-term transients behind the invasion front, a phenomenon known as dynamical stabilisation. We combine absolute stability theory and computation to predict how the width of the stabilised region depends on parameter values. We develop our calculations in the context of a model for a cyclic predator-prey system, in which the invasion front and spatiotemporal oscillations of predators and prey are separated by a region in which the coexistence steady state is dynamically stabilised.
We consider a seasonally forced SIR epidemic model where periodicity occurs in the contact rate. This periodical forcing represents successions of school terms and holidays. The epidemic dynamics are described by a switched system. Numerical studies in such a model have shown the existence of periodic solutions. First, we analytically prove the existence of an invariant domain [Formula: see text] containing all periodic (harmonic and subharmonic) orbits. Then, using different scales in time and variables, we rewrite the SIR model as a slow-fast dynamical system and we establish the existence of a macroscopic attractor domain [Formula: see text], included in [Formula: see text], for the switched dynamics. The existence of a unique harmonic solution is also proved for any value of the magnitude of the seasonal forcing term which can be interpreted as an annual infection. Subharmonic solutions can be seen as epidemic outbreaks. Our theoretical results allow us to exhibit quantitative characteristics about epidemics, such as the maximal period between major outbreaks and maximal prevalence.
By hybridization and backcrossing, alleles can surmount species boundaries and be incorporated into the genome of a related species. This introgression of genes is of particular evolutionary relevance if it involves the transfer of adaptations between populations. However, any beneficial allele will typically be associated with other alien alleles that are often deleterious and hamper the introgression process. In order to describe the introgression of an adaptive allele, we set up a stochastic model with an explicit genetic makeup of linked and unlinked deleterious alleles. Based on the theory of reducible multitype branching processes, we derive a recursive expression for the establishment probability of the beneficial allele after a single hybridization event. We furthermore study the probability that slightly deleterious alleles hitchhike to fixation. The key to the analysis is a split of the process into a stochastic phase in which the advantageous alleles establishes and a deterministic phase in which it sweeps to fixation. We thereafter apply the theory to a set of biologically relevant scenarios such as introgression in the presence of many unlinked or few closely linked deleterious alleles. A comparison to computer simulations shows that the approximations work well over a large parameter range.
Carey’s Equality pertaining to stationary models is well known. In this paper, we have stated and proved a fundamental theorem related to the formation of this Equality. This theorem will provide an in-depth understanding of the role of each captive subject, and their corresponding follow-up duration in a stationary population. We have demonstrated a numerical example of a captive cohort and the survival pattern of medfly populations. These results can be adopted to understand age-structure and aging process in stationary and non-stationary population models.
The mutation-selection process is the most fundamental mechanism of evolution. In 1935, R. A. Fisher proved his fundamental theorem of natural selection, providing a model in which the rate of change of mean fitness is equal to the genetic variance of a species. Fisher did not include mutations in his model, but believed that mutations would provide a continual supply of variance resulting in perpetual increase in mean fitness, thus providing a foundation for neo-Darwinian theory. In this paper we re-examine Fisher’s Theorem, showing that because it disregards mutations, and because it is invalid beyond one instant in time, it has limited biological relevance. We build a differential equations model from Fisher’s first principles with mutations added, and prove a revised theorem showing the rate of change in mean fitness is equal to genetic variance plus a mutational effects term. We refer to our revised theorem as the fundamental theorem of natural selection with mutations. Our expanded theorem, and our associated analyses (analytic computation, numerical simulation, and visualization), provide a clearer understanding of the mutation-selection process, and allow application of biologically realistic parameters such as mutational effects. The expanded theorem has biological implications significantly different from what Fisher had envisioned.
Identifying the critical domain size necessary for a population to persist is an important question in ecology. Both demographic and environmental stochasticity impact a population’s ability to persist. Here we explore ways of including this variability. We study populations with distinct dispersal and sedentary stages, which have traditionally been modelled using a deterministic integrodifference equation (IDE) framework. Individual-based models (IBMs) are the most intuitive stochastic analogues to IDEs but yield few analytic insights. We explore two alternate approaches; one is a scaling up to the population level using the Central Limit Theorem, and the other a variation on both Galton-Watson branching processes and branching processes in random environments. These branching process models closely approximate the IBM and yield insight into the factors determining the critical domain size for a given population subject to stochasticity.
In this work, we develop a data-driven modelling framework to reproduce the locomotion of fish in a confined environment. Specifically, we highlight the primary characteristics of the motion of individual zebrafish (Danio rerio), and study how these can be suitably encapsulated within a mathematical framework utilising a limited number of calibrated model parameters. Using data captured from individual zebrafish via automated visual tracking, we develop a model using stochastic differential equations and describe fish as a self propelled particle moving in a plane. Based on recent experimental evidence of the importance of speed regulation in social behaviour, we extend stochastic models of fish locomotion by introducing experimentally-derived processes describing dynamic speed regulation. Salient metrics are defined which are then used to calibrate key parameters of coupled stochastic differential equations, describing both speed and angular speed of swimming fish. The effects of external constraints are also included, based on experimentally observed responses. Understanding the spontaneous dynamics of zebrafish using a bottom-up, purely data-driven approach is expected to yield a modelling framework for quantitative investigation of individual behaviour in the presence of various external constraints or biological assays.
In simple SI epidemic and endemic models, three classes of incidence functions are identified for their potential to be associated with host extinction: weakly upper density-dependent incidences are never associated with host extinction. Power incidences that depend on the number of susceptibles and infectives by powers strictly between 0 and 1 are associated with initial-constellation-dependent host extinction for all parameter values. Homogeneous incidences, of which frequency-dependent incidence is a very particular case, and power incidences are associated with global host extinction for certain parameter constellations and with host survival for others. Laboratory infection experiments with salamander larvae are equally well fitted by power incidences and certain upper density-dependent incidences such as the negative binomial incidence and do not rule out homogeneous incidences such as an asymmetric frequency-dependent incidence either.