# Mathematical Modelling In Ecology And Evolution Pdf Chapter 12 Evolutionary Invasion

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Klanjscek T. Dynamic energy budgets and bioaccumulation: a model for marine mammals and marine mammal populations. Caswell H, Neubert MG. Reactivity and transient dynamics of discrete-time ecological systems. Journal of Difference Equations and Applications ;11

## Conceptual Frameworks and Methods for Advancing Invasion Ecology

Evolutionary game theory EGT is the application of game theory to evolving populations in biology. It defines a framework of contests, strategies, and analytics into which Darwinian competition can be modelled. Price 's formalisation of contests, analysed as strategies, and the mathematical criteria that can be used to predict the results of competing strategies.

Evolutionary game theory differs from classical game theory in focusing more on the dynamics of strategy change. Evolutionary game theory has helped to explain the basis of altruistic behaviours in Darwinian evolution. It has in turn become of interest to economists , sociologists , anthropologists , and philosophers. Classical non-cooperative game theory was conceived by John von Neumann to determine optimal strategies in competitions between adversaries.

A contest involves players, all of whom have a choice of moves. Games can be a single round or repetitive. The approach a player takes in making his moves constitutes his strategy.

Rules govern the outcome for the moves taken by the players, and outcomes produce payoffs for the players; rules and resulting payoffs can be expressed as decision trees or in a payoff matrix. Classical theory requires the players to make rational choices. Each player must consider the strategic analysis that his opponents are making to make his own choice of moves.

Evolutionary game theory started with the problem of how to explain ritualized animal behaviour in a conflict situation; "why are animals so 'gentlemanly or ladylike' in contests for resources? John Maynard Smith considered that incompatible with Darwinian thought, [6] where selection occurs at an individual level, so self-interest is rewarded while seeking the common good is not.

Maynard Smith, a mathematical biologist, turned to game theory as suggested by George Price, though Richard Lewontin 's attempts to use the theory had failed. Maynard Smith realised that an evolutionary version of game theory does not require players to act rationally—only that they have a strategy. The results of a game shows how good that strategy was, just as evolution tests alternative strategies for the ability to survive and reproduce. In biology, strategies are genetically inherited traits that control an individual's action, analogous with computer programs.

The success of a strategy is determined by how good the strategy is in the presence of competing strategies including itself , and of the frequency with which those strategies are used. Participants aim to produce as many replicas of themselves as they can, and the payoff is in units of fitness relative worth in being able to reproduce. It is always a multi-player game with many competitors.

Rules include replicator dynamics, in other words how the fitter players will spawn more replicas of themselves into the population and how the less fit will be culled , in a replicator equation. The replicator dynamics models heredity but not mutation, and assumes asexual reproduction for the sake of simplicity. Games are run repetitively with no terminating conditions. Results include the dynamics of changes in the population, the success of strategies, and any equilibrium states reached.

Unlike in classical game theory, players do not choose their strategy and cannot change it: they are born with a strategy and their offspring inherit that same strategy. Evolutionary game theory encompasses Darwinian evolution, including competition the game , natural selection replicator dynamics , and heredity. Evolutionary game theory has contributed to the understanding of group selection , sexual selection , altruism , parental care , co-evolution , and ecological dynamics. Many counter-intuitive situations in these areas have been put on a firm mathematical footing by the use of these models.

The common way to study the evolutionary dynamics in games is through replicator equations. These show the growth rate of the proportion of organisms using a certain strategy and that rate is equal to the difference between the average payoff of that strategy and the average payoff of the population as a whole. The attractors stable fixed points of the equations are equivalent with evolutionarily stable states.

A strategy which can survive all "mutant" strategies is considered evolutionarily stable. In the context of animal behavior, this usually means such strategies are programmed and heavily influenced by genetics , thus making any player or organism's strategy determined by these biological factors. Evolutionary games are mathematical objects with different rules, payoffs, and mathematical behaviours. Each "game" represents different problems that organisms have to deal with, and the strategies they might adopt to survive and reproduce.

Evolutionary games are often given colourful names and cover stories which describe the general situation of a particular game. Representative games include hawk-dove , [1] war of attrition , [15] stag hunt , producer-scrounger , tragedy of the commons , and prisoner's dilemma. Strategies for these games include hawk, dove, bourgeois, prober, defector, assessor, and retaliator. The various strategies compete under the particular game's rules, and the mathematics are used to determine the results and behaviours.

The first game that Maynard Smith analysed is the classic hawk dove [a] game. It was conceived to analyse Lorenz and Tinbergen's problem, a contest over a shareable resource. The contestants can be either a hawk or a dove. These are two subtypes or morphs of one species with different strategies. The hawk first displays aggression, then escalates into a fight until it either wins or is injured loses.

The dove first displays aggression, but if faced with major escalation runs for safety. If not faced with such escalation, the dove attempts to share the resource. Given that the resource is given the value V, the damage from losing a fight is given cost C: [1]. The actual payoff however depends on the probability of meeting a hawk or dove, which in turn is a representation of the percentage of hawks and doves in the population when a particular contest takes place.

That in turn is determined by the results of all of the previous contests. The population regresses to this equilibrium point if any new hawks or doves make a temporary perturbation in the population. The solution of the hawk dove game explains why most animal contests involve only ritual fighting behaviours in contests rather than outright battles.

The result does not at all depend on " good of the species " behaviours as suggested by Lorenz, but solely on the implication of actions of so-called selfish genes. In the hawk dove game the resource is shareable, which gives payoffs to both doves meeting in a pairwise contest. Where the resource is not shareable, but an alternative resource might be available by backing off and trying elsewhere, pure hawk or dove strategies are less effective. If an unshareable resource is combined with a high cost of losing a contest injury or possible death both hawk and dove payoffs are further diminished.

A safer strategy of lower cost display, bluffing and waiting to win, is then viable — a bluffer strategy. The game then becomes one of accumulating costs, either the costs of displaying or the costs of prolonged unresolved engagement. It is effectively an auction; the winner is the contestant who will swallow the greater cost while the loser gets the same cost as the winner but no resource.

This is because in the war of attrition any strategy that is unwavering and predictable is unstable, because it will ultimately be displaced by a mutant strategy which relies on the fact that it can best the existing predictable strategy by investing an extra small delta of waiting resource to ensure that it wins.

Therefore, only a random unpredictable strategy can maintain itself in a population of bluffers. The contestants in effect choose an acceptable cost to be incurred related to the value of the resource being sought, effectively making a random bid as part of a mixed strategy a strategy where a contestant has several, or even many, possible actions in their strategy.

This implements a distribution of bids for a resource of specific value V, where the bid for any specific contest is chosen at random from that distribution. The result is that the cumulative population of quitters for any particular cost m in this "mixed strategy" solution is:.

The intuitive sense that greater values of resource sought leads to greater waiting times is borne out. This is observed in nature, as in male dung flies contesting for mating sites, where the timing of disengagement in contests is as predicted by evolutionary theory mathematics.

In the war of attrition there must be nothing that signals the size of a bid to an opponent, otherwise the opponent can use the cue in an effective counter-strategy.

There is however a mutant strategy which can better a bluffer in the war of attrition game if a suitable asymmetry exists, the bourgeois strategy. Bourgeois uses an asymmetry of some sort to break the deadlock. In nature one such asymmetry is possession of a resource.

The strategy is to play a hawk if in possession of the resource, but to display then retreat if not in possession. This requires greater cognitive capability than hawk, but bourgeois is common in many animal contests, such as in contests among mantis shrimps and among speckled wood butterflies.

Games like hawk dove and war of attrition represent pure competition between individuals and have no attendant social elements.

Where social influences apply, competitors have four possible alternatives for strategic interaction. This is shown on the adjacent figure, where a plus sign represents a benefit and a minus sign represents a cost. At first glance it may appear that the contestants of evolutionary games are the individuals present in each generation who directly participate in the game. But individuals live only through one game cycle, and instead it is the strategies that really contest with one another over the duration of these many-generation games.

So it is ultimately genes that play out a full contest — selfish genes of strategy. The contesting genes are present in an individual and to a degree in all of the individual's kin.

This can sometimes profoundly affect which strategies survive, especially with issues of cooperation and defection. William Hamilton , [21] known for his theory of kin selection , explored many of these cases using game-theoretic models. Kin-related treatment of game contests [22] helps to explain many aspects of the behaviour of social insects , the altruistic behaviour in parent-offspring interactions, mutual protection behaviours, and co-operative care of offspring.

For such games, Hamilton defined an extended form of fitness — inclusive fitness , which includes an individual's offspring as well as any offspring equivalents found in kin. If individual a i sacrifices their "own average equivalent fitness of 1" by accepting a fitness cost C, and then to "get that loss back", w i must still be 1 or greater than Hamilton went beyond kin relatedness to work with Robert Axelrod , analysing games of co-operation under conditions not involving kin where reciprocal altruism came into play.

Eusocial insect workers forfeit reproductive rights to their queen. It has been suggested that kin selection, based on the genetic makeup of these workers, may predispose them to altruistic behaviours.

This explanation of insect eusociality has, however, been challenged by a few highly-noted evolutionary game theorists Nowak and Wilson [26] who have published a controversial alternative game theoretic explanation based on a sequential development and group selection effects proposed for these insect species.

A difficulty of the theory of evolution, recognised by Darwin himself, was the problem of altruism. If the basis for selection is at an individual level, altruism makes no sense at all.

But universal selection at the group level for the good of the species, not the individual fails to pass the test of the mathematics of game theory and is certainly not the general case in nature. The solution to this problem can be found in the application of evolutionary game theory to the prisoner's dilemma game — a game which tests the payoffs of cooperating or in defecting from cooperation.

It is the most studied game in all of game theory. The analysis of the prisoner's dilemma is as a repetitive game. This affords competitors the possibility of retaliating for defection in previous rounds of the game. Many strategies have been tested; the best competitive strategies are general cooperation, with a reserved retaliatory response if necessary. The pay-off for any single round of the game is defined by the pay-off matrix for a single round game shown in bar chart 1 below.

In multi-round games the different choices — co-operate or defect — can be made in any particular round, resulting in a certain round payoff.

## The biology and mathematical modelling of glioma invasion: a review

Bolker, and David J. Estimating initial epidemic growth rates. Bulletin of Mathematical Biology , pages , Interspecific dominance via vocal interactions mediates altitudinal zonation in neotropical singing mice. A method for detecting positive growth autocorrelation without marking individuals. Testing an autonomous acoustic telemetry positioning system for fine-scale space use in marine animals. Methods in Ecology and Evolution , 4 6 , June

We give a very short introduction to discrete and continuum models for the evolutionary and spatial dynamics of cancer through two case studies: a model for the evolutionary dynamics of cancer cells under cytotoxic therapy and a model for the mechanical interaction between healthy and cancer cells during tumour growth. First we develop the discrete models, whereby the dynamics of single cells are described through a set of rules that result in branching random walks. Then we present the corresponding continuum models, which are formulated in terms of non-local and nonlinear partial differential equations, and we summarise the key properties of their solutions. Finally, we carry out numerical simulations of the discrete models and we construct numerical solutions of the corresponding continuum models. The biological implications of the results obtained are briefly discussed.

One idea that comes to mind is that the sex ratio is even because many species have sex chromosomes, which segregate among the gametes that determine sex e. Example: Interlocus sexually antagonistic selection in water striders Our final example introduces a new feature, in which we suppose that individuals of different classes exhibit This is another example of evolutionary branching, where a population evolves toward a point at which a polymorphism arises Geritz et al. When the proportion of double matings is high enough, males that invest little in accessory proteins can cheat by taking advantage of the accessory proteins provided by other males, saving their re This phenomenon, termed indirect selection, has played an important role in many evolutionary models see, e.

## A Biologist's Guide to Mathematical Modeling in Ecology and Evolution

Persistent genetic variation within populations presents an evolutionary problem, as natural selection and genetic drift tend to erode genetic diversity. Models of balancing selection were developed to account for the maintenance of genetic variation observed in natural populations. Negative frequency-dependent selection is a powerful type of balancing selection that maintains many natural polymorphisms, but it is also commonly misinterpreted.

Evolutionary game theory EGT is the application of game theory to evolving populations in biology. It defines a framework of contests, strategies, and analytics into which Darwinian competition can be modelled. Price 's formalisation of contests, analysed as strategies, and the mathematical criteria that can be used to predict the results of competing strategies.

#### Perspective ARTICLE

Adult gliomas are aggressive brain tumours associated with low patient survival rates and limited life expectancy. The most important hallmark of this type of tumour is its invasive behaviour, characterized by a markedly phenotypic plasticity, infiltrative tumour morphologies and the ability of malignant progression from low- to high-grade tumour types. Indeed, the widespread infiltration of healthy brain tissue by glioma cells is largely responsible for poor prognosis and the difficulty of finding curative therapies. Meanwhile, mathematical models have been established to analyse potential mechanisms of glioma invasion. In this review, we start with a brief introduction to current biological knowledge about glioma invasion, and then critically review and highlight future challenges for mathematical models of glioma invasion. Gliomas are the most common primary tumours of the central nervous system CNS in adults.

Not a MyNAP member yet? Register for a free account to start saving and receiving special member only perks. Science is a particular way of knowing about the world. In science, explanations are restricted to those that can be inferred from confirmable data—the results obtained through observations and experiments that can be substantiated by other scientists. Anything that can be observed or measured is amenable to scientific investigation. Explanations that cannot be based on empirical evidence are not a part of science. The history of life on earth is a fascinating subject that can be studied through observations made today, and these observations have led to compelling accounts of how organisms have changed over time.

Most evolutionary thinking is based on the notion of fitness and related ideas such as fitness landscapes and evolutionary optima. Nevertheless, it is often unclear what fitness actually is, and its meaning often depends on the context. Here we argue that fitness should not be a basal ingredient in verbal or mathematical descriptions of evolution. Instead, we propose that evolutionary birth-death processes, in which individuals give birth and die at ever-changing rates, should be the basis of evolutionary theory, because such processes capture the fundamental events that generate evolutionary dynamics. In evolutionary birth-death processes, fitness is at best a derived quantity, and owing to the potential complexity of such processes, there is no guarantee that there is a simple scalar, such as fitness, that would describe long-term evolutionary outcomes. We discuss how evolutionary birth-death processes can provide useful perspectives on a number of central issues in evolution. What is evolution?

Handbook of the Mathematics of the Arts and Sciences pp Cite as. Invasive species are nonindigenous plants and animals that have the potential to cause great harm to both the environment and native species. If an invasive species is able to survive and spread throughout the environment, there may be large financial losses to the public. Public policy makers and scientists are responsible for developing and funding programs to control and eradicate invasive species. These programs are founded on the science of invasion ecology and the biological characteristics of the invader.

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