Biological Population Model Research Articles

Ecosystem Evolution is About Variation and Persistence, not Populations and Reproduction

Building upon a non-standard understanding of evolutionary process focusing on variation and persistence, I will argue that communities and ecosystems can evolve by natural selection as emergent individuals. Evolutionary biology has relied ever increasingly on the modeling of population dynamics. Most have taken for granted that we all agree on what is a population. Recent work has reexamined this perceived consensus. I will argue that there are good reasons to restrict the term “population” to collections of monophyletically related replicators and interactors, which explains why many existing models in population biology exclude by definition many genuine evolving biological individuals such as communities and ecosystems. By studying a case of community evolution (a symbiotic termite–fungus community), we will see that it is variation that is important to evolutionary processes, not populations. Variation within a population is only one of many types of variation that can lead to evolution by natural selection. The upshot of focusing on variation is that cases of community and ecosystem adaptive change become tractable in evolutionary terms. I will show that complex emergent individuals such as communities and ecosystems cannot be fully accommodated by conventional population/reproduction models but can be accommodated by variation/persistence models.

Two-dimensional time fractional-order biological population model and its analytical solution

In this article, a mathematical model has been developed for the generalized time fractional–order biological population model (GTFBPM). The fractional derivative has been described in the Caputo sense. The model has been solved by a recent approximate analytic method so called the fractional reduced differential transform method (FRDTM). Using this method, it is possible to find the exact solution as well as closed approximate solution of a differential equation. Three numerical examples of GTFBPM have been provided in order to check the effectiveness, accuracy and convergence of the method. The special advantage of using this computational technique is that it is very easy to implement and takes small size of computation contrary to other numerical methods while dealing complex and tedious physical problems arising in various branches of natural sciences and engineering.

Quadratic model of inter-population interaction: Investigation of stability areas

A development quadratic model of the heterogeneous biological population consisting of a few sub-populations is investigated. The classic stabilization problem of solutions for the system of ordinary quadratic differential equations, which describing this model, is solved. Examples are given.

Replicator Equations and Models of Biological Populations and Communities

An overview of a general approach for mathematical modeling of evolving heterogeneous populations using a wide class of selection systems and replicator equations (RE) is presented. The method allows visualizing evolutionary trajectories of evolving heterogeneous populations over time, while still enabling use of analytical tools of bifurcation theory. The developed theory involves introducing escort systems of auxiliary “keystone" variables, which reduce complex multi-dimensional inhomogeneous models to low dimensional systems of ODEs that in many cases can be investigated analytically. In addition to a comprehensive theoretical framework, a set of examples of the method’s applicability to questions ranging from preventing the tragedy of the commons to cancer therapy is presented.

Гладкие модели биологических популяций

Гладкие модели биологических популяций

Modeling of biological population problems using the element-free kp-Ritz method

A degenerate parabolic equation arising in the spatial diffusion of biological population is analyzed using the mesh-free kp-Ritz method. The mesh-free kernel particle estimate is employed to approximate the 2D displacement field. A system of discrete equations is obtained through application of the Ritz minimization procedure to the energy expressions. To validate the accuracy of the results and stability of the present method, convergence studies were carried out based on influences of support size and number of nodes. Effectiveness of the mesh-free kp-Ritz method for biological population model is investigated by numerical examples in this paper. The present results were compared with results reported in extant literature and were found to be in good agreement with the literature.

Stochastic models in population biology: from dynamic noise to Bayesian description and model comparison for given data sets

We study some stochastic models from population biology, namely epidemiology, which can be solved analytically. From the solution of the stochastic processes we construct the likelihood function and compare the maximum likelihood parameter estimation procedure with the Bayesian approach of obtaining an explicit probability for the parameters given the available data. Finally, we compare one model against the other in the Bayesian framework, both models performing on the same simulated data set. In some cases of data obtained under one model with specific parameter values, the model comparison favours the model not underlying the simulated data. This apparently paradoxical situation arises in parameter regions which do not easily give sufficient information to the simulated data to reject simpler models.

Fractional Riccati Equation Rational Expansion Method For Fractional Differential Equations

In this paper, a new fractional Riccati equation rational expansion method is proposed to establish new exact solutions for fractional differential equations. For illustrating the validity of this metho d, we apply it to the nonlinear fractional Sharma-Tasso- Olever (STO) equation, the nonlinear time fractional biological population model and the nonlinear fractional foam drainage equation. Compared with the existing results in the literature, more exact solutions are o btained by the proposed method. We also illustrate the application of the established exact solutions.

On the optimal harvesting of size-structured population dynamics

This work is concerned with a kind of optimal control problem for a size-structured biological population model. Well-posedness of the state system and an adjoint system are proved by means of Banach’s fixed point theorem. Existence and uniqueness of optimal control are shown by functional analytical approach. Optimality conditions describing the optimal strategy are established via tangent and normal cones technique. The results are of the first ones for this novel structure.

And So It Grows: Using a Computer-Based Simulation of a Population Growth Model to Integrate Biology & Mathematics

We provide a 5E structured-inquiry lesson so that students can learn more of the mathematics behind the logistic model of population biology. By using models and mathematics, students understand how population dynamics can be influenced by relatively simple changes in the environment.

A New Approach for Solving Fractional Partial Differential Equations

We propose a new approach for solving fractional partial differential equations based on a nonlinear fractional complex transformation and the general Riccati equation and apply it to solve the nonlinear time fractional biological population model and the (4+1)-dimensional space-time fractional Fokas equation. As a result, some new exact solutions for them are obtained. This approach can be suitable for solving fractional partial differential equations with more general forms than the method proposed by S. Zhang and H.-Q. Zhang (2011).

Lie algebra method for solving biological population model

In this paper, the Lie algebraic method is applied to solve biological population models described by time-inhomogeneous birth-death processes. Notwithstanding no obvious symmetry, the solution is expressed by matrix exponentials through suitably generated low-dimensional Lie algebras. This methodology may offer useful insights for other biological and ecological applications.

Existence and uniqueness of steady state solutions of a nonlocal diffusive logistic equation

In this paper, we consider a dynamical model of population biology which is of the classical Fisher type, but the competition interaction between individuals is nonlocal. The existence, uniqueness, and stability of the steady state solution of the nonlocal problem on a bounded interval with homogeneous Dirichlet boundary conditions are studied.

Traveling waves in discrete models of biological populations with sessile stages

A common approach to describing invasions of non-native species into previously unoccupied habitat is to consider the speed of population expansion and the existence of traveling waves. Typical existence theorems for traveling waves require some compactness properties of the next-generation operator. Many realistic modeling assumptions, however, give rise to non-compact operators; for example the occurrence of sessile stages during the life cycle of an individual. Recent results have extended the existence theory of traveling waves to a large class of weakly compact operators, but conditions can be difficult to check and not easily accessible to theoretical ecologists. In this paper, we give a new proof for the existence of traveling waves in a large class of equations where the next generation operator is not compact, but rather the sum of an integral operator and a contraction. We illustrate our proof with a model for the dispersal of a plant species with a seed-bank and a model for dispersal of stream insects with larval stages.

(Flat)fish stocks in an ecosystem and evolutionary perspective

The delineation of natural populations has many faces. While a stock relates to a management unit of organisms, the biological concept of a population may relate to a demographic/ecological perspective on interacting organisms or an evolving group of organisms. In addition, it has become increasingly clear that the time and spatial scales of the ecological and evolutionary population have much in common. Evolutionary population models, which are the focus of this paper, harbor independent information on census population size, population dynamics, population history and population connectivity. Fish populations continuously adapt to the changing environmental conditions. However, the impact of and response to fishing, climate and pollution stress are disconnected in time and lead to measurable changes in the genomes. Several new insights have emerged lately, such as the limited but biologically meaningful subtle genetic differentiation, the contribution of population connectivity, the consequences of the very small effective population sizes and the interaction between environment and evolution. This leads to the importance of management of the genetic monitoring of populations, the inclusion of adaptation in management models and the contribution of marine protected areas to guarantee the long term integrity of marine ecosystems. Of immediate significance is the match between stocks and the biological population model.

Integrating mechanistic organism–environment interactions into the basic theory of community and evolutionary ecology

This paper presents an overview of how mechanistic knowledge of organism-environment interactions, including biomechanical interactions of heat, mass and momentum transfer, can be integrated into basic theoretical population biology through mechanistic functional responses that quantitatively describe how organisms respond to their physical environment. Integrating such functional responses into simple community and microevolutionary models allows scaling up of the organism-level understanding from biomechanics both ecologically and temporally. For community models, Holling-type functional responses for predator-prey interactions provide a classic example of the functional response affecting qualitative model dynamics, and recent efforts are expanding analogous models to incorporate environmental influences such as temperature. For evolutionary models, mechanistic functional responses dependent on the environment can serve as fitness functions in both quantitative genetic and game theoretic frameworks, especially those concerning function-valued traits. I present a novel comparison of a mechanistic fitness function based on thermal performance curves to a commonly used generic fitness function, which quantitatively differ in their predictions for response to environmental change. A variety of examples illustrate how mechanistic functional responses enhance model connections to biologically relevant traits and processes as well as environmental conditions and therefore have the potential to link theoretical and empirical studies. Sensitivity analysis of such models can provide biologically relevant insight into which parameters and processes are important to community and evolutionary responses to environmental change such as climate change, which can inform conservation management aimed at protecting response capacity. Overall, the distillation of detailed knowledge or organism-environment interactions into mechanistic functional responses in simple population biology models provides a framework for integrating biomechanics and ecology that allows both tractability and generality.

Homotopy Analysis Method for Solving Biological Population Model

In this paper, the homotopy analysis method (HAM) is applied to solve generalized biological population models. The fractional derivatives are described by Caputo's sense. The method introduces a significant improvement in this field over existing techniques. Results obtained using the scheme presented here agree well with the analytical solutions and the numerical results presented in Ref. [6]. However, the fundamental solutions of these equations still exhibit useful scaling properties that make them attractive for applications.

Constructing ecologies

We synthesize the generic properties of ecologically realistic multi-trophic level models and define criteria for ecological realism. We define an “ecospace” in which all ecologically realistic dynamics are confined, and construct “resource rays” that define the resources available to each species at every point in the ecospace. Resource rays for a species are lines from a vertex of maximum resource to the opposite boundary where no resources are available. The growth functions of all biota normally decrease along their resource rays, and change sign from positive to negative. This property prescribes that each species must have a zero isosurface within the ecospace. We illustrate our conditions on a highly cited three trophic level model from population dynamics, showing how to extend this system biologically consistently to a closed ecological system. Our synthesis extends the concept of carrying capacity of population models to explicitly include exhaustion of limiting resources, and so allows for population biology models to be considered as ecologically closed systems with respect to a key limiting nutrient. This approach unifies many theoretical and applied models in a common biogeochemical framework, facilitates better understanding of the key structures of complex ecologies, and suggests strategies for efficient design of experiments.

Clarifying Baker's Law

Baker's Law states that colonization by self-compatible organisms is more likely to be successful than colonization by self-incompatible organisms because of the ability for self-compatible organisms to produce offspring without pollination agents. This simple model has proved very successful in plant ecology and has been applied to various contexts, including colonizing or ruderal species, islands colonizers, invasive species or mating system variation across distribution ranges. Moreover, it is one of the only models in population biology linking two traits of major importance in ecology, namely dispersal and mating system. Although Baker's Law has stimulated a large number of empirical studies reporting the association of self-fertilization and colonizing ability in various contexts, the data have not established a general pattern for the association of traits. In this paper, a critical position is adopted to discuss and clarify Baker's Law. From the literature referring to Baker's Law, an analysis made regarding how mating success is considered in such studies and discrepancies with population genetics theory of mating systems are highlighted. The data reporting the association of self-fertilization and colonizing ability are also briefly reviewed and the potential bias in interpretation is discussed. Lastly, a recent theoretical model analysing the link between colonizing ability and self-fertilization is considered. Evolutionary predictions are actually more complex than Baker's intuitive arguments. It appears that Baker's Law encompasses a variety of ecological scenarios, which cannot be considered a priori as equivalent. Questioning what has been considered as self-evident for more than 50 years seems a reasonable objective to analyse in-depth dispersal and mating system traits.

A robustness analysis of biological population models with protection zone

Establishing protection zone has been widely used to preserve endangered species. We study the effect of protection zone in random environment by a stochastic model. Our results show that the establishment of nature reserves can protect endangered species effectively in good conditions. Under very bad conditions, though protection zone cannot protect species from extinction, it also can slow down the speed of extinction. Based on our results, we give some methods to strengthen the effect of protection zone and provide a method of dividing the protection zone. In addition, we introduce some examples and use several numerical simulations to enhance our main results.