Darwinian Evolution, Hypercycles and Game Theory

Abstract

Adaptation and evolution are quasi synonymous in popular language and Darwinian evolution is a prime application of complex adaptive system theory. We will see that adaptation does not happen automatically and discuss the concept of “error catastrophe” as a possible root for the downfall of a species. Venturing briefly into the mysteries surrounding the origin of life, we will investigate the possible advent of a “quasispecies” in terms of mutually supporting hypercycles. The basic theory of evolution is furthermore closely related to game theory, the mathematical theory of interacting agents, viz of rationally acting economic persons.

Exercises

6.1.1 The One-Dimensional Ising Model

Solve the one-dimensional Ising model

$$H \ =\ J\sum_i s_i s_{i+1}\,+\,B\sum_i s_i$$

by the transfer matrix method presented in Sect. 6.3.2 and calculate the free energy $F(T,\!B)$, the magnetization $M(T,B)$ and the susceptibility $\chi(\!T)= \lim_{B\to0}\frac{\partial M(T,B)}{\partial B}$.

6.1.2 Error Catastrophe

For the prebiotic quasispecies model Eq. (6.51) consider tower-like autocatalytic reproduction rates $W_{jj}$ and mutation rates $W_{ij}$ ($i\ne j$) of the form

$$W_{ii} \ =\ \left\{ \begin{array}{cc} 1 & i=1\\ 1-\sigma & i>1 \end{array} \right. , \qquad W_{ij} \ =\ \left\{ \begin{array}{cl} u_+ & i=j+1\\ u_- & i=j-1\\ 0 & i\ne j \ \textrm{otherwise} \end{array} \right. ,$$

with $\sigma,u_\pm\in[0,1]$. Determine the error catastrophe for the two cases $u_+=u_-\equiv u$ and $u_+=u$, $u_-=0$. Compare it to the results for the tower landscape discussed in Sect. 6.3.3.

Hint: For the stationary eigenvalue equation (6.51), with $\dot x_i=0$ ($i=1,\ldots$), write $x_{j+1}$ as a function of x _j and $x_{j-1}$. This two-step recursion relation leads to a $2\times 2$ matrix. Consider the eigenvalues/vectors of this matrix, the initial condition for x ₁, and the normalization condition $\sum_i x_i<\infty$ valid in the adapting regime.

6.1.3 Models of Life

Go to the Internet, e.g. http://cmol.nbi.dk/javaapp.php, and try a few JAVA applets simulating models of life. Select a model of your choice and study the literature given.

6.1.4 Competition for Resources

The competition for scarce resources has been modelled in the quasispecies theory, see Eq. (6.48), by an overall constraint on population density. With

$$\dot x_i \ =\ W_{ii} x_i \qquad W_{ii} \ =\ f r_i-d, \qquad \dot f \ =\ a-f \sum_i r_i x_i$$

((6.64))

one models the competition for the resource f explicitly, with a $(fr_i)$ being the regeneration rate of the resource f (species i) and d the mortality rate. Eq. (6.64) does not contain mutation terms $\sim W_{ij}$ describing a simple ecosystem. Which is the steady-state value of the total population density $C= \sum_i x_i$ and of the resource level f? Is the ecosystem stable?

6.1.5 Hypercycles

Consider the reaction equations (6.52) and (6.53) for $N=2$ molecules and a homogeneous network. Find the fixpoints and discuss their stability.

6.1.6 The Prisoner’s Dilemma on a Lattice

Consider the stability of intruders in the prisoner’s dilemma Eq. (6.62) on a square lattice, as the one illustrated in Fig. 6.10. Namely, the case of just one and of two adjacent defectors/cooperators in a background of cooperators/defectors. Who survives?

6.1.7 Nash Equilibrium

Examine the Nash equilibrium and its optimality for the following two-player game:

Each player acts either cautiously or riskily. A player acting cautiously always receives a low pay-off. A player playing riskily gets a high pay-off if the other player also takes a risk. Otherwise, the risk-taker obtains no reward.