Koella (2000). The spatial spread of altruism

Ref: Koella, J. C. (2000). The spatial spread of altruism versus the evolutionary response of egoists. Proc. R. Soc. Lond. B, 267, 1979-1985. 

General Model: 
Individuals interact on a toroidal square lattice. Sites can be empty or occupied by an individual with investment level I, between 0 and 0.99. Can mutuate and change I by plus/minus 0.01. Individuals interact with all neighbors within interaction distance.
Birthrate is given by
\[ b(I) = b_0 + \beta (1 - e^{-k\bar{I}}) - \gamma I \]
where $$\bar{I}$$ is the mean level of investment within the agent's interaction distance (among his N neighbors. Offspring are successful only if they find an empty site within the parent's dispersal distance. When competition for sites arises, empty sites are occupied by the offspring of the parent with the locally highest probability of reproduction.
All sites are initialized with 50x50 egoist (I = 0) individuals, and death rate is 0.1 for all individuals.

Results of General Model:
Confirm results of previous studies showing that limited dispersal enables clusters of altruists to evolve. No intermediate values of altruism. Increased dispersal distance for all individuals decreases average investment, and requires higher initial benefits of altruism (k) to be maximized.

Coevolution of Dispersal Distance
Dispersal distance, d, was allowed to evolve as well. Interaction distance was held constant at 1. There was a cost associated with dispersal, so the reproduction rate was:
\[ b(I) = b_0 + \beta (1 - e^{-k\bar{I}}) - \gamma I  - \delta d \]

Results:
The level of altruism that evolved was much lower. Also, altruists tended to have very low dispersal rates, and egoists tended to have much greater dispersal distances.

Coevolution of Interaction Distance
Dispersal distance was held constant at 1, and interaction distance was allowed to evolve. No cost was associated with interaction distance, so the first reproduction function was used.

Results:
As in the first condition, altruists and egoists evolved with no intermediate values. Altruists tended to have small interaction distances, egoists tended to have large interaction distances. This also led to the evolution of spatial patterns - stripes of alternating altruists and egoists.

Conclusions:

1. Investment into altruism can evolve to a high level under rather general conditions.
2. Egoists cannot be eliminated from the population, but respond to the spread of altruism by increasing their dispersal distance and the distance over which they interact with neighbors.
3. The combined evolution of investment into altruism and the interaction distance can lead to stable spatial patterns. 

Notes: 

• No change in cost in the number of interactions, as with many other models. 
• The average investment level is used, so that sparse vs. dense populations are not taken into account.
• Individuals disperse at reproduction, but do not accumulate wealth or move.