Modelling Food Webs

We review theoretical approaches to the understanding of food webs. After an overview of the available food web data, we discuss three different classes of models. The first class comprise static models, which assign links between species according to some simple rule. The second class are dynamical models, which include the population dynamics of several interacting species. We focus on the question of the stability of such webs. The third class are species assembly models and evolutionary models, which build webs starting from a few species by adding new species through a process of "invasion" (assembly models) or "speciation" (evolutionary models). Evolutionary models are found to be capable of building large stable webs.Comment: 34 pages, 2 figures. To be published in "Handbook of graphs and networks" S. Bornholdt and H. G. Schuster (eds) (Wiley-VCH, Berlin

Evolutionary ecology in-silico: Does mathematical modelling help in understanding the "generic" trends?

Motivated by the results of recent laboratory experiments (Yoshida et al. Nature, 424, 303-306 (2003)) as well as many earlier field observations that evolutionary changes can take place in ecosystems over relatively short ecological time scales, several ``unified'' mathematical models of evolutionary ecology have been developed over the last few years with the aim of describing the statistical properties of data related to the evolution of ecosystems. Moreover, because of the availability of sufficiently fast computers, it has become possible to carry out detailed computer simulations of these models. For the sake of completeness and to put these recent developments in the proper perspective, we begin with a brief summary of some older models of ecological phenomena and evolutionary processes. However, the main aim of this article is to review critically these ``unified'' models, particularly those published in the physics literature, in simple language that makes the new theories accessible to wider audience.Comment: 28 pages, LATEX, 4 eps figure

Coevolutionary dynamics of a variant of the cyclic Lotka-Volterra model with three-agent interactions

We study a variant of the cyclic Lotka-Volterra model with three-agent interactions. Inspired by a multiplayer variation of the Rock-Paper-Scissors game, the model describes an ideal ecosystem in which cyclic competition among three species develops through cooperative predation. Its rate equations in a well-mixed environment display a degenerate Hopf bifurcation, occurring as reactions involving two predators plus one prey have the same rate as reactions involving two preys plus one predator. We estimate the magnitude of the stochastic noise at the bifurcation point, where finite size effects turn neutrally stable orbits into erratically diverging trajectories. In particular, we compare analytic predictions for the extinction probability, derived in the Fokker-Planck approximation, with numerical simulations based on the Gillespie stochastic algorithm. We then extend the analysis of the phase portrait to heterogeneous rates. In a well-mixed environment, we observe a continuum of degenerate Hopf bifurcations, generalizing the above one. Neutral stability ensues from a complex equilibrium between different reactions. Remarkably, on a two-dimensional lattice, all bifurcations disappear as a consequence of the spatial locality of the interactions. In the second part of the paper, we investigate the effects of mobility in a lattice metapopulation model with patches hosting several agents. We find that strategies propagate along the arms of rotating spirals, as they usually do in models of cyclic dominance. We observe propagation instabilities in the regime of large wavelengths. We also examine three-agent interactions inducing nonlinear diffusion.Comment: 22 pages, 13 figures. v2: version accepted for publication in EPJ

Deleting species from model food webs

We use food webs generated by a model to investigate the effects of deleting species on other species in the web and on the web as a whole. The model incorporates a realistic population dynamics, adaptive foragers and other features which allow for the construction of model webs which resemble empirical food webs. A large number of simulations were carried out to produce a substantial number of model webs on which deletion experiments could be performed. We deleted each species in four hundred distinct model webs and determined, on average, how many species were eliminated from the web as a result. Typically only a small number of species became extinct; in no instance was the web close to collapse. Next, we examined how the the probability of extinction of a species depended on its relationship with the deleted species. This involved the exploration of the concept of indirect predator and prey species and the extent that the probability of extinction depended on the trophic level of the two species. The effect of deletions on the web itself was studied by searching for keystone species, whose removal caused a major restructuring of the community, and also by looking at the correlation between a number of food web properties (number of species, linkage density, fraction of omnivores, degree of cycling and redundancy) and the stability of the web to deletions. With the exception of redundancy, we found little or no correlation. In particular, we found no evidence that complexity in terms of increased species number or links per species is destabilising.Comment: 30 pages, 9 figure

Modeling the Disappearance of the Neanderthals Using Concepts of Population Dynamics and Ecology

Current hypotheses regarding the disappearance of Neanderthals (NEA) in Europe fall into two main categories: climate change, and competition. Here we review current research and existing mathematical models that deal with this question, and we propose an approach that incorporates and permits the investigation of the current hypotheses. We have developed a set of differential equations that model population dynamics of anatomically modern humans (AMH) and NEA, their ecological relations to prey species, and their mutual interactions. The model allows investigators to explore each of the two main categories or combinations of both, as well as various forms of competition and/or interference within the context of competition. The model is designed to include a wide variety of hypotheses and associated archaeological evidence, not focused on a particular hypothesis regarding NEA extinction. It therefore provides investigators with a model to impartially examine various hypotheses (individually or in combination) regarding climatic effects, differential resource use, differences in birth/death rates and carrying capacities, competition, interference, disease, interbreeding, and cultural distinctions that might have led to the extinction of NEA. Moreover, the model accommodates the design of scenarios concerning—for example—population growth, hunting, competitive interactions, cultural differences, and climatic influences to investigate which concepts best explain the rapid disappearance of NEA. In addition, our model is a modification of the classical Lotka-Volterra model for a wide range of any two populations competing for a common resource. Specifically, our model explicitly includes the resource as an additional variable, a dependence of important population parameters on resource, as well as accommodates treating one of the populations as invasive

Fixed-to-mobile substitution in the US, EU, and China: Forecasting technology diffusion using the Lotka-Volterra Competition model

Objectives The first purpose of this thesis is to test the performance of the Lotka – Volterra Competition model in forecasting demand for technologies. Secondly, the paper aims to determine the interrelationship between the markets and their expected behaviors based on population theories. Thirdly, it attempts to gauge the similarities and differences of market behaviors in the most developed economies based on GDPpc as of October 2018. Summary Total annual subscription for each market was used to perform in-sample forecasts. Parameterization was obtained using the Gauss-Newton non-linear least squares method with the Marquardt algorithm. Then, the stable equilibria were shown in the interactive outcome graphs, which indicate that the theoretical suggestions are well-supported by historical market patterns. Conclusions The results indicate high fitting performance (R-squares>0.98) with estimated data close to that of actual observations. Despite data complications, the model has a good degree of accuracy. The competitive relationships for the US, the EU, and China are suggested to be amensalism, amensalism, and pure competition, respectively. The equilibrium analyses show that in all scenarios, the mobile cellular market dominates the fixed-line phone market. Over time, mobile phones will substitute fixed – line phones and obtain maximum growth

The Role of Behavioral Dynamics in Determining the Patch Distributions of Interacting Species

The effect of the behavioral dynamics of movement on the population dynamics of interacting species in multipatch systems is studied. The behavioral dynamics of habitat choice used in a range of previous models are reviewed. There is very limited empirical evidence for distinguishing between these different models, but they differ in important ways, and many lack properties that would guarantee stability of an ideal free distribution in a single-species system. The importance of finding out more about movement dynamics in multispecies systems is shown by an analysis of the effect of movement rules on the dynamics of a particular two-species–two-patch model of competition, where the population dynamical equilibrium in the absence of movement is often not a behavioral equilibrium in the presence of adaptive movement. The population dynamics of this system are explored for several different movement rules and different parameter values, producing a variety of outcomes. Other systems of interacting species that may lack a dynamically stable distribution among patches are discussed, and it is argued that such systems are not rare. The sensitivity of community properties to individual movement behavior in this and earlier studies argues that there is a great need for empirical investigation to determine the applicability of different models of the behavioral dynamics of habitat selection

Bayesian Experimental Design For Bayesian Hierarchical Models With Differential Equations For Ecological Applications

Ecologists are interested in the composition of species in various ecosystems. Studying population dynamics can assist environmental managers in making better decisions for the environment. Traditionally, the sampling of species has been recorded on a regular time frequency. However, sampling can be an expensive process due to financial and physical constraints. In some cases the environments are threatening, and ecologists prefer to limit their time collecting data in the field. Rather than convenience sampling, a statistical approach is introduced to improve data collection methods for ecologists by studying the dynamics associated with populations of interest. Population models including the logistic equation and the Lotka-Volterra differential equations are employed to simulate species composition. This research focuses on sequentially learning about the behavior of dynamical systems to better inform ecologists of when to sample. The developed algorithm of sequential optimality designs sampling regimes to assist ecologists with resource allocation while providing maximum information from the data. This research in its entirety constructs a method for designing sampling schedules for ecologists based on the dynamics associated with temporal ecological models