3. Ecological Dependency
The ecological state eco\Theta_{eco}eco --- including prey abundance, prey trait distribution, and environmental factors --- dynamically shapes both benefits and costs. For instance:
In environments with slower prey, the marginal value of extreme aerodynamic refinement diminishes.
In high-altitude habitats, respiratory capacity (T2T_2T2) becomes disproportionately valuable.
When prey evolve evasive maneuvers, predator traits (T3,T4T_3, T_4T3,T4) must coevolve to maintain relative fitness.
Thus, the fitness function is not fixed but co-constructed by predator and prey, embodying the CAS principle that adaptive landscapes are dynamic rather than static.
C. Replicator--Mutator Dynamics and Agent-Based Representation
To capture the evolutionary dynamics generated by the genotype--phenotype--fitness mapping, we require a formalism that accommodates both selection and variation. The replicator--mutator equation provides such a framework at the population level, while agent-based models (ABMs) allow explicit simulation of individual variation and ecological interaction. Together, these approaches embody the dual requirements of rigor and realism in a CAS framework.
1. Replicator--Mutator Formalism
Let xi(t)x_i(t)xi(t) denote the frequency of genotype iii at generation ttt, with associated phenotype Ti\mathbf{T}_iTi and fitness wi(t)w_i(t)wi(t). The frequency update rule is:
xi(t+1)=1w(t)jxj(t)wj(t)Qji,x_i(t+1) \;=\; \frac{1}{\bar{w}(t)} \sum_j x_j(t) \, w_j(t) \, Q_{ji},xi(t+1)=w(t)1jxj(t)wj(t)Qji,
where:
QjiQ_{ji}Qji is the mutation--recombination transition probability, giving the likelihood that genotype jjj produces genotype iii as offspring.
w(t)=jxj(t)wj(t)\bar{w}(t) = \sum_j x_j(t) w_j(t)w(t)=jxj(t)wj(t) is the mean population fitness, ensuring normalization.
This equation captures both replication proportional to fitness and variation through mutation and recombination. In the absence of mutation (Qji=ijQ_{ji} = \delta_{ij}Qji=ij), the equation reduces to the classical replicator dynamic. With mutation, the population explores genotype space, avoiding permanent fixation on suboptimal local peaks.
When combined with the fitness function defined in Section III.B, the replicator--mutator equation yields a nonlinear dynamical system in which population trajectories can converge to stable equilibria (adaptive attractors), oscillate due to coevolutionary feedback, or bifurcate under parameter shifts.
