Matrix Models for Population, Disease, and Evolutionary Dynamics

In physics, there is a growing interest in studying stochastic processes described by evolution equations such as nonlinear master equations and nonlinear Fokker–Planck equations that define the so-called nonlinear Markov processes and are nonlinear with respect to probability densities. In this context, however, relatively little is known about nonlinear Markov processes defined by nonlinear Markov chains. In the present work, we demonstrate explicitly how the nonlinear Markov chain approach can be carried out by addressing a model for evolutionary population dynamics. In line with the nonlinear Markov chain approach, we derive a measure that tells us how attractive it is for a biological entity to evolve towards a particular biological type. Likewise, a measure for the noise level of the evolutionary process is obtained. Both measures are found to be implicitly time dependent. Finally, a simulation scheme for the many-body system corresponding to the Markov chain model is discussed.

We give a direct proof of well-posedness of solutions to general selection-mutation and structured population models with measures as initial data. This is motivated by the fact that some stationary states of these models are measures and not $L^1$ functions, so the measures are a more natural space to study their dynamics. Our techniques are based on distances between measures appearing in optimal transport and common arguments involving Picard iterations. These tools provide a simplification of previous approaches and are applicable or adaptable to a wide variety of models in population dynamics.

Population Dynamics: Mathematic Models of Population, Development, and Natural Resources

The present note studies some cross-diffusion models in population dynamics. The well-posedness of these systems is established and Lyapunov functions are introduced. Furthermore, the given results imply the well-posedness of some filters in signal enhancement processes of one-dimensional signals.

Background:The geographic range of the tick Amblyomma americanum, a vector of diseases of public health significance such as ehrlichiosis, has expanded from the southeast of the United States northward during the 20th century. Recently, populations of this tick have been reported to be present close to the Canadian border in Michigan and New York states, but established populations are not known in Canada. Previous research suggests that changing temperature patterns with climate change may influence tick life cycles and permit northward range expansion of ticks in the northern hemisphere.Objectives:We aimed to estimate minimal temperature conditions for survival of A. americanum populations at the northern edge of the tick’s range and to investigate the possibility of range expansion of A. americanum into northern U.S. states and southern Canada in the coming decades.Methods:A simulation model of the tick A. americanum was used, via simulations using climate data from meteorological stations in the United States and Canada, to estimate minimal temperature conditions for survival of A. americanum populations at the northern edge of the tick’s range.Results:The predicted geographic scope of temperature suitability [ annual cumulative degree days (DD) ] included most of the central and eastern U.S. states east of longitude 110°W, which is consistent with current surveillance data for the presence of the tick in this region, as well as parts of southern Quebec and Ontario in Canada. Regional climate model output raises the possibility of northward range expansion into all provinces of Canada from Alberta to Newfoundland and Labrador during the coming decades, with the greatest northward range expansion (up to by the year 2100) occurring under the greenhouse gas (GHG) emissions of Representative Concentration Pathway (RCP) 8.5. Predicted northward range expansion was reduced by approximately half under the reduced GHG emissions of RCP4.5.Discussion:Our results raise the possibility of range expansion of A. americanum into northern U.S. states and southern Canada in the coming decades, and conclude that surveillance for this tick, and the diseases it transmits, would be prudent. https://doi.org/10.1289/EHP5668

Population models provide insights into population dynamics under diverse and untested chemical exposure scenarios, supporting their environmental risk assessment (ERA). In this study, we investigate the interplay of temperature and imidacloprid exposure on population dynamics using an Individual-Based Model (IBM) incorporating a dynamic energy budget (DEB) model for population dynamics and toxicokinetic-toxicodynamic models of the General Unified Threshold model for Survival (GUTS) framework to predict toxicity effects. For this, we tested different model configurations, where i) only the DEB parameters are corrected for temperature, as is common practice, and ii) also the TKTD parameters of the GUTS model are corrected for temperature. In doing so, we aim to evaluate the importance of temperature corrections in the GUTS model within an IBM framework. As expected, increased temperature amplitudes increase the range of simulated population sizes, and chemical exposure reduces the maximum population size. The combined effect of correcting both the DEB and TKTD parameters, however, yield an overall strongly negative effect on population sizes, particularly at lower temperatures. These results highlight the necessity of temperature-sensitive parameterization in population models for a protective risk assessment under the projected future climate conditions with increased temperatures and variability. Future considerations include incorporating local adaptations and acclimatization, particularly in different climate zones, to accurately interpret population model outcomes in the context of evolving environmental conditions. Such insights contribute to the refinement of ecological realism in ERA, enhancing the robustness of chemical risk management strategies.

Historical events have been used to explain a wide range of phenomena including geographical distributions of species, community diversity, and population structure. At the level of individuals, historical effects in which past conditions influence future performance are particularly likely to occur in long-lived organisms that store resources between seasons and that form organs months or years before their elaboration. Such carry-over mechanisms have been documented in several perennial plant species, but the implications for population processes are poorly known. In this study, I examine how the history of individuals influences their future performance, population dynamics, and life cycle structure in the long-lived herb Lathyrus vernus. Overall effects of plant history on population dynamics, in terms of growth rate, reproductive values, stable stage distribution, and elasticities, are examined by comparing an ordinary first-order matrix model with a second-order matrix model. In the latter, not only the present state of individuals, but also their past state is allowed to influence future fate. The results demonstrate that the history of individuals is sometimes important in models of population dynamics. Plant size change over a one-year period was negatively correlated among time intervals. Addition of the previous year's stage in population models shifted the growth rate from positive (λ = 1.010) to negative (λ = 0.986) and increased the proportion of small established individuals in the stable stage distribution. If historical effects are due to a capacity to buffer environmental variation and regain size or state, as in L. vernus, then recruitment contributes less and stasis more to population growth than suggested by ahistorical models. The presence of historical effects at the level of individuals, in any form, may have important consequences for population development and should be included in any interpretation of the life-cycle structure.

Integrated modelling of insect population dynamics at two temporal scales

Nonparametric spline based models of stage structured population dynamics provide an effective way of decomposing observed population dynamics into birth and death rate processes. Currently available methodology only considers simple life histories in which all individuals pass through the same sequence of stages in the same way. This paper presents techniques for obtaining birth and death rates from structured population time series of organisms with complex life histories. For example, in many species males and females develop at the same rate in early stages, but differently in late stages. The models are motivated by the need to obtain birth and death rates for a pest subject to biocontrol by a parasitoid.

Summary Population dynamic modelling often entails parameterizing quite sophisticated biological and ecological mechanisms. For models of moderate mechanistic complexity, this has traditionally been done in an ad hoc manner, with different parameters being estimated independently. The point estimates so obtained are then used for model simulation, perhaps with some further ad hoc adjustment based on comparison with any available data on population dynamics. Quantitative assessments of model adequacy and prediction uncertainty are not easily made using this approach. As an alternative, the paper investigates the practical feasibility of fitting a moderately complex population dynamic model directly and simultaneously to all the data available for parameterization of the model, and to all available data on the population dynamics of the target animal. This alternative approach allows us to combine all available quantitative information on the target species, to assess the viability of the model, the mutual consistency of model and different sources of data and to estimate the uncertainties that are associated with model-based predictions. The target organism in this study is the freshwater amphipod Gammarus pulex (L.), which we model using a stage-structured population dynamic model, implemented via a set of delay differential equations describing the basic demography of the population. Target data include population dynamic data from two sites, information on basic physiological relationships and environmental temperature data. Fitting is performed by using a non-linear least squares approach supplemented with a bootstrapping method for avoiding small scale local minima in the least squares objective function. Variance estimation is performed by further bootstrapping. Interest in Gammarus pulex population dynamics in this case is primarily related to likely population level responses to chemical stressors, and for this we examine predicted ‘recovery times’ following exposure to a known toxicant.

This article studies nonlinear n-resource-consumer autonomous system with age-structured consumer population. The model of consumer population dynamics is described by a delayed transport equation, and the dynamics of resource patches are described by ODE with saturated intake rate. The delay models the digestion period of generalist consumer and is included in the calorie intake rate, which impacts the consumer’s fertility and mortality. Saturated intake rate models the inhibition effect from the behavioral change of the resource patches when they react to the consumer population growing or from the crowding effect of the consumer. The conditions for the existence of trivial, semi-trivial, and non-trivial equilibria and their local asymptotic stability were obtained. The local asymptotic stability/instability of non-trivial equilibrium of a system with depleted patches is defined by new derived criteria, which relate the demographic characteristics of consumers with their search rate, growth rate of resource in patches, and behavioral change of the food resource when consumer population grows. The digestion period of a generalist consumer does not cause local asymptotical instabilities of consumer population at the semi-trivial and nontrivial equilibria. These theoretical results may be used in the study of metapopulation dynamics, desert locust populations dynamics, prey-predator interactions in fisheries, etc. The paper uses numerical experiments to confirm and illustrate all dynamical regimes of the n-resource-consumer population.

BackgroundIt is often assumed that evolution takes place on very large timescales. Countering this assumption, rapid evolutionary dynamics are increasingly documented in biological systems, e.g. in the context of predator–prey interactions, species coexistence and invasion. It has also been shown that rapid evolution can facilitate the evolution of cooperation. In this context often evolutionary dynamics influence population dynamics, but in spatial models rapid evolutionary dynamics also emerge with constant population sizes. Currently it is not clear how well these spatial models apply to species in which individuals are not embedded in fixed spatial structures. To address this issue we employ an agent-based model of group living individuals. We investigate how positive assortment between cooperators and defectors and pay-off differences between cooperators and defectors depend on the occurrence of evolutionary dynamics.ResultsWe find that positive assortment and pay-off differences between cooperators and defectors differ when comparing scenarios with and without selection, which indicates that rapid evolutionary dynamics are occurring in the selection scenarios. Specifically, rapid evolution occurs because changes in positive assortment feed back on evolutionary dynamics, which crucially impacts the evolution of cooperation. At high frequencies of cooperators these feedback dynamics increase positive assortment facilitating the evolution of cooperation. In contrast, at low frequencies of cooperators rapid evolutionary dynamics lead to a decrease in assortment, which acts against the evolution of cooperation. The contrasting dynamics at low and high frequencies of cooperators create positive frequency-dependent selection.ConclusionsRapid evolutionary dynamics can influence the evolution of cooperation in group-living species and lead to positive frequency-dependent selection even if population size and maximum group-size are not affected by evolutionary dynamics. Rapid evolutionary dynamics can emerge in this case because sufficiently strong selective pressures allow evolutionary and demographic dynamics, and consequently also feedback between assortment and evolution, to occur on the same timescale. In particular, emerging positive frequency-dependent selection could be an important explanation for differences in cooperative behaviors among different species with similar population structures such as humans and chimpanzees.

We study a model of stochastic evolutionary game dynamics in which the probabilities that agents choose suboptimal actions are dependent on payoff consequences. We prove a sample path large deviation principle, characterizing the rate of decay of the probability that the sample path of the evolutionary process lies in a prespecified set as the population size approaches infinity. We use these results to describe excursion rates and stationary distribution asymptotics in settings where the mean dynamic admits a globally attracting state, and we compute these rates explicitly for the case of logit choice in potential games.

  Stochastic evolutionary game dynamics in hierarchy are very popular in economics and in management sciences. This study extends the model of stochastic evolutionary game dynamics. This paper highlights hierarchy selection model and we show that the hierarchy selection games are more stable without structures in this paper. Furthermore, an example in industrial organization is given to rationally explain the theory in this work.   Key words: Fitness, fixation probability, invasion, game theory, economics, hierarchy, evolutionary game.

Mathematical modelling is an important tool in population dynamics that allows describing various biotechnological and ecological processes. The classical models (e.g., the Monod-type models, modelling microbial growth or the Rosenzweig-MacArthur predator-prey model, among others) assume a per capita growth rate that is linear w.r.t. the food intake and a constant per capita death rate. In the present work, we consider generic birth and death rates based on several observations proposed by A.J. Terry. For sufficiently small values of the consumption, the reproduction will be zero rather than linearly increasing. Also, there will always be a limit to the rate at which an individual can reproduce. On the other hand, every organism needs some minimal energy intake, in order to survive. Therefore, under this threshold the death rate should be higher. Using the idea of non-linear birth and death rates, we propose models that are generalizations of classical models in population dynamics. We study the asymptotic properties of the solutions of the considered models. We show that in some cases those models have much richer dynamics and, thus, allow us to model more complex behaviour of the biological system. Also, we investigate their ability to model real processes by comparing the results of the parametric identification process to experimental data for different microbial cultures.