Evolutionary Games and Periodic Fitness
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Abstract
One thing that nearly all stability concepts in evolutionary game theory have in common is that they use a timeindependent fitness matrix. Although this is a reasonable assumption for mathematical purposes, in many situations in real life it seems to be too restrictive. We present a model of an evolutionary game, driven by replicator dynamics, where the fitness matrix is a variable rather than a constant, i.e., the fitness matrix is timedependent. In particular, by considering periodically changing fitness matrices, we model seasonal effects in evolutionary games. We discuss a model with a continuously changing fitness matrix as well as a related model in which the changes occur periodically at discrete points in time. A numerical analysis shows stability of the periodic orbits that are observed. Moreover, trajectories leading to these orbits from arbitrary starting points synchronize their motion in time. Several examples are discussed.
Keywords
Evolutionary game Replicator dynamics Periodic fitness1 Introduction
Evolutionary game theory, started by the work of Maynard Smith and Price [7], studies the dynamic development of populations. Here, a population consists of interacting individuals of finitely many different types. Interactions between different types lead to different fitnesses for these types (e.g., number of offspring). Consequentially, the population distribution, i.e., the relative frequency of each of the types is subject to change over time. In evolutionary models, the rate and the direction of this change are determined by the dynamics. There are several dynamics in use (cf. Hofbauer & Sigmund [6]), the most common one being the replicator dynamics (Taylor & Jonker [10]) and the best response dynamics (Gilboa & Matsui [5]).
One thing that all dynamics in evolutionary games have in common is that they make use of timeindependent fitness matrices. Although this is a reasonable assumption for mathematical purposes, in many situations in real life it seems to be too restrictive. For instance, if different types require different resources, then a possible environmental effect of having one type abundantly present could be that its resources run low, which in turn would lead to a lower fitness. Also, there may be external effects that influence the fitnesses of the different types in different ways. Little research has been done in this area; one example being the work by Broom [2], who discusses evolutionary games, where the fitness matrix converges to a fixed limit matrix.
In this paper, we present a model of an evolutionary game, driven by replicator dynamics, where the fitness matrix is a variable rather than a constant, i.e., the fitness matrix is timedependent, in the following way: We introduce periodically changing fitness matrices to model seasonal effects in evolutionary games. We present, by means of an example, the model of an evolutionary game with periodic fitnesses, i.e., the fitness matrix continuously changes in time in a periodic fashion. In the example, the population distribution is shown to converge to a periodic orbit. Moreover, trajectories leading to this orbit from arbitrary starting points synchronize their motion in time. Furthermore, we discuss games in which the fitness matrix still changes periodically, but only at discrete points in time. A similar result as for the continuously changing fitness matrix is obtained for an example where only two fitness matrices are used alternatingly over fixed time intervals.
2 The Model
Both the original ESSdefinition and the replicator dynamics assume that the fitness matrix is constant. In this paper, we investigate the replicator dynamics and a type of stability of the population distribution in evolutionary games with a timedependent fitness matrix A(t). We will call this periodic stability. We do so by means of the following example. This example is based on the idea of having three types, two of which have a fitness that periodically depends on time, sometimes doing very good, sometimes very bad, while for another type the fitness is not directly affected by time at all:
Example 1
 (i)
σ determines the size or amplitude of the variation. Notice that if σ=0, then the fitness matrix is timeindependent.
 (ii)
ρ determines the time it takes to complete one period. The smaller the value of ρ, the more time the population has to adapt to the changing environment.
 (iii)
α is introduced to control the (timeindependent) fitness of the type 2 individuals.
Proposition 1
In the evolutionary game corresponding to the timedependent fitness matrix A(t) from Example 1, for parameter values α=0.88, ρ=0.1 and σ=1, the process {x(t):t≥0} converges to a periodically stable orbit.
Moreover, trajectories leading to this orbit from arbitrary starting points, synchronize their motion in time.^{1}
3 Analysis
 Case 1:

α<0.75. Then, for any value of t, the unique ESS of the game with fitness matrix A(t) combines types 1 and 3, while type 2 goes extinct. The ESS’s oscillate in periods of \(\frac{2\pi}{\rho}\) between \(\mathbf{p}=(\frac{3}{4}, 0, \frac{1}{4})\) when \(t=k\cdot\frac{2\pi}{\rho}\) and \(\mathbf{p}=(\frac {1}{4}, 0, \frac{3}{4})\) when \(t=\frac{\pi}{\rho}+k\cdot\frac{2\pi}{\rho }\). Furthermore, at any time t the replicator dynamics drives the population distribution in the direction of the current ESS at time t.
 Case 2:

α>1. Now, at any time there is a unique ESS, namely e _{2}=(0,1,0), and indeed the dynamics converge to e _{2}. This has to do with the fact that, when α>1, the fitness of type 2 individuals against any population distribution \(\tilde{\mathbf{x}}\) that does not contain any individuals of type 2 is \(\mathbf{e}_{2}A\tilde{\mathbf{x}}^{\mathrm{T}}=\alpha>1\geq\tilde{\mathbf {x}}A\tilde{\mathbf{x}}^{\mathrm{T}}\).
 Case 3:

0.75<α<1. Now, depending on t, the process will be attracted to a point on the line segment between \((\frac{1}{4}, 0, \frac{3}{4})\) and \((\frac {3}{4}, 0, \frac{1}{4})\) or to e _{2}. However, some care is needed, for in this case e _{2} is no ESS as will be explained below. These timedependent attractors allow for the possibility for all types to survive and some kind limit cycle to occur. One can observe that at certain time intervals, namely when \(\alpha>1\frac {1}{4}\cos^{2}(\rho t)\), the type 2 individuals have the upper hand, whereas at the other time intervals types 1 and 3 flourish.
The value of ρ also plays an important role in the overall process. One can see ρ as the rate for the population to adapt to changing fitnesses. If ρ is high, then the population hardly has any time to adapt. In this case, the entire process behaves as if there was just one fitness matrix, the time average one. At the opposite, if ρ is small, the population has lots of time to adapt. In the extreme case where ρ approaches zero, some types may go extinct before their fitness values recover. The value ρ=0.1 turns out to be good to observe a cyclic pattern.
3.1 Continuously Changing Fitness Matrix
When we examine the process obtained by the replicator dynamics on the continuously changing fitness matrix A(t), then we observe that no matter where we start in the interior of the simplex, the process always converges to the same cyclic trajectory. This is illustrated in Fig. 1. Again the parameters used are: α=0.88, ρ=0.1, and σ=1.
In order to show that the observed periodic orbit is indeed periodically stable we used the tool Ariadne (cf. Collins et al. [3]) that was developed for the analysis of dynamic systems using rigorous numerical methods. (See the Appendix for an overview of rigorous numerics and the methods used for the analysis.) We considered the replicator dynamics for the specific parameter values given, which has a forcing period of \(T=\frac{2\pi}{\rho}\). An approximation to the timeT return map r over an initial domain D was computed, along with an error bound in the uniform norm. The interval Newton operator was then used to prove the existence of a fixedpoint of r, yielding a periodT orbit of the replicator dynamics.
One of the eigenvalues of the derivative matrix of the return map was computed to be approximately 0.95, whereas the other two were calculated to be of order 10^{−15}, which is comparable to the machine epsilon. This validates the observation made during the simulations, that in the region of interest, there is very fast convergence to an invariant manifold, followed by slow convergence to the fixedpoint of the return map. The eigenvalue 0.95 of the derivative matrix indicates that a perturbation of the system changes the position of the fixed point of the return map by a factor of 20 times the magnitude of the perturbation. In particular, the return map needed to be computed to high accuracy in order to prove the existence of the periodic solution.
3.2 Changes in Fitness at Periodic Discrete Points in Time
4 Other Patterns with Periodic Fitness Matrices
5 Conclusions
In conclusion, we have discovered that it is possible to reach a periodically stable population pattern when following the replicator dynamics based on a periodically changing fitness matrix. Because the fitness matrix changes continuously in time, the force field of the dynamics that makes the population move into the direction of a particular population distribution, changes with it. In our stepwise periodic model, we have sharp changes of the fitness matrix at fixed times, and the number of attraction points is finite. In fact, for the particular case examined, there is just one attraction point all the time, but the combined process never reaches it.
We want to remark that we could only employ the Ariadne tool for showing stability of the orbits observed, because the fitness matrix was changing in a deterministic way. Such would not have been possible when it was changing in stochastic way, because then there would not necessarily be any closed loop.
A challenging question is to predict the existence of periodically stable orbits based on the periodic fitness matrix. A second question is to find closed form expressions for the periodic orbits observed.
Movies for the dynamic processes discussed can be observed at http://www.youtube.com/watch?v=C65Z7fcLA4s.
Footnotes
 1.
Movies for the dynamic processes discussed can be observed at http://www.youtube.com/watch?v=C65Z7fcLA4s.
Notes
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