Jonathan T. Rowell

The Territorial Raider game and graph derangements

A derangement of a graph G = ( V , E ) is an injective function f : V ź V such that for all v ź V , f ( v ) ź v and ( v , f ( v ) ) ź E . Not all graphs admit a derangement and previous results have characterized graphs with derangements using neighborhood conditions for subsets of V . We establish an alternative criterion for the existence of derangements on a graph. We analyze strict Nash equilibria of the biologically motivated Territorial Raider game, a multi-player competition for resources in a spatially structured population based on animal raiding and defending behavior. We find that a graph G admits a derangement if and only if there is a strict Nash equilibrium of the Territorial Raider game on G .

Cooperation in finite populations: Being alone helps

Abstract We consider the evolution of cooperation in finite populations and we model a scenario where two individuals can interact only if both intend to do so with their counterpart. This feature allows a possibility for individuals to remain alone for a given round and not interact with anybody. Such an individual receives a baseline payoff rather than one based upon a matrix game. We provide sufficient conditions on the payoff matrix that will guarantee fixation probabilities to be monotone relative to the baseline payoff. We then apply the findings to the Prisoner’s Dilemma and Hawk-Dove games. In both cases, the possibility that an individual might remain alone increases the chances that cooperation or non-aggression fixes within the population. Moreover, weak selection models overlap with our model, and we consider how one can generalize our model even further.

Cooperative behaviour in theory and practice: leading undergraduate research in behaviour mathematical biology

The US National Science Foundation has promoted the early integration of undergraduate students into academic research environments by funding activities such as the Research Experiences for Undergraduates (REU) programs. Here, we discuss the operation of the first year for one REU site held on the campus of the University of North Carolina at Greensboro. Eight students from across the country were brought together for a 10-week summer programme in mathematical biology to work on research topics related to the establishment of cooperative behaviour. The students were paired in four teams, which approached the dynamic tension between cooperation and defection from a variety of mathematical and contextual perspectives. Two projects employed agent-based models on lattice environments: the first consisting of sparse populations of mobile individuals and the second featuring age-structured populations, history-dependent fitness, and the possibility of kinship recognition. A third project looked at the evolutionary dynamics of behavioural frequencies in a model of society comprised of three social strata. The final project focused on the ecological range distribution of non-cooperative, cooperative and kleptoparasitic populations in heterogeneous resource environments. We describe both the general results of the students’ research efforts and our observations on the efficacy of several enrichment activities provided to the students over the course of the REU summer.

The evolution of cooperation is affected by the persistence of fitness effects, the neighborhood size and their interaction

Evolutionary game theory and the Prisoner’s Dilemma (PD) Game in particular have been used to study the evolution of cooperation. We consider a population of asexually reproducing, age-structured individuals in a two-dimensional square-lattice structure. The individuals employ fixed cooperative or defecting strategy towards their neighbours in repeating interactions to accumulate reproductive fitness. We focus on the effects of the persistence of past interactions and interactive neighbourhood size on the evolution of cooperation. We show that larger neighbourhood sizes are generally detrimental to cooperation and that the persistence of fitness effects decreases the likelihood of the evolution of cooperation in small neighbourhoods. However, for larger neighbourhood sizes the persistence effect is reversed. Thus, our study corroborates earlier studies that population structure increases the evolutionary potential for cooperative behaviour in a PD paradigm. This finding may explain the heterogeneity of previous results on the effect of neighbourhood size and cautions that the persistence of fitness outcomes needs to be considered in analyses of the evolution of cooperative behaviour. The persistence of fitness outcomes of pairwise interactions may vary dramatically in biological and social systems and could have profound effects on the evolution of cooperation in various contexts.

Game Theoretical Model of Cancer Dynamics with Four Cell Phenotypes

The development of a cancerous tumor requires affected cells to collectively display an assortment of characteristic behaviors that contribute differently to its growth. A heterogeneous population of tumor cells is far more resistant to treatment than a homogeneous one as different cell types respond dissimilarly to treatments; yet, these cell types are also in competition with one another. This paper models heterogeneous cancer cell interactions within the tumor mass through several game theoretic approaches including classical normal form games, replicator dynamics, and spatial games. Our concept model community consists of four cell strategies: an angiogenesis-factor-producing cell, a proliferative cell, a cytotoxin producing cell, and a neutral stromal cell. By comparing pairwise strategic interactions, invasibility and counter-invasibility, we establish conditions for dominance and the existence of both monomorphic and polymorphic equilibria. The spatial game supports co-occurrence among multiple subpopulations in accordance with biological observations of developing tumors. As the tumor progresses from primarily stromal cells to a more malignant state, angiogenic and cytotoxic cells form clusters while proliferative cells are widespread. The clustering of certain subpopulations suggests insight into the behaviors of cancer cells that could influence future treatment strategies.

Dynamical facilitation of the ideal free distribution in nonideal populations

Abstract The ideal free distribution (IFD) requires that individuals can accurately perceive density‐dependent habitat quality, while failure to discern quality differences below a given perception threshold results in distributions approaching spatial uniformity. Here, we investigate the role of population growth in restoring a nonideal population to the IFD. We place a simple model of discrete patch choice under limits to the resolution by which patch quality is perceived and include population growth driven by that underlying quality. Our model follows the populations distribution through both breeding and dispersal seasons when perception limits differ in their likely influence. We demonstrate that populations of perception limited movers can approximate an IFD provided sufficient population growth; however, the emergent IFD would be temporally inconstant and correspond to reproductive events. The time to emergence of the IFD during breeding is shorter under exponential growth than under logistic growth. The IFD during early colonization of a community persists longer when more patches are available to individuals. As the population matures and dispersal becomes increasingly random, there is an oscillation in the observance of IFD, with peaks most closely approximating the IFD occurring immediately after reproductive events, and higher reproductive rates producing distributions closer to the IFD.

A Game-Theoretic Model of Cholera with Optimal Personal Protection Strategies

Cholera is an acute gastro-intestinal infection that affects millions of people throughout the world each year, primarily but not exclusively in developing countries. Because of its public health ramifications, considerable mathematical attention has been paid to the disease. Here we consider one neglected aspect of combating cholera: personal participation in anti-cholera interventions. We construct a game-theoretic model of cholera in which individuals choose whether to participate in either vaccination or clean water consumption programs under assumed costs. We find that relying upon individual compliance significantly lowers the incidence of the disease as long as the cost of intervention is sufficiently low, but does not eliminate it. The relative costs of the measures determined whether a population preferentially adopts a single preventative measure or employs the measure with the strongest early adoption.

A continuous ideal free distribution approach to the dynamics of selfish, cooperative and kleptoparasitic populations

Population distributions depend upon the aggregate behavioural responses of individuals to a range of environmental factors. We extend a model of ideally motivated populations to describe the local and regional consequences of interactions between three populations distinguished by their levels of cooperation and exploitation. Inspired by the classic prisoners dilemma game, stereotypical fitness functions describe a baseline non-cooperative population whose per capita fitness decreases with density, obligate co-operators who initially benefit from the presence of conspecifics, and kleptoparasites who require heterospecifics to extract resources from the environment. We examine these populations in multiple combinations, determine where both local and regional coexistence is permitted, and investigate conditions under which one population will invade another. When they invade co-operators in resource-rich areas, kleptoparasites initiate a dynamic instability that leads to the loss of both populations; however, selfish hosts, who can persist at low densities, are immune to this risk. Furthermore, adaptive movement may delay the onset of instability as dispersal relieves dynamic stress. Selfish and cooperative populations default to mutual exclusion, but asymmetric variations in interference strength may relax this condition and permit limited sympatry within the environment. Distinct sub-communities characterize the overall spatial structure.