2. Agent-Based Representation
While replicator--mutator equations provide analytical tractability, they assume infinite population sizes and well-mixed interactions. To relax these assumptions and incorporate ecological heterogeneity, we employ agent-based modeling (ABM).
In ABM, each predator and prey is represented as an explicit agent characterized by:
A genotype vector g\mathbf{g}g.
A phenotype vector T\mathbf{T}T derived via the mapping in Section III.A.
A fitness value determined by predatory encounters (Section III.B).
The ABM proceeds in discrete time steps:
1. Trait Expression: each agent computes T\mathbf{T}T from its genotype.
2. Interaction Phase: predators attempt hunts against prey, with success probability phuntp_{hunt}phunt. Prey that evade survive to reproduce.
3. Reproduction Phase: surviving agents reproduce in proportion to their fitness, subject to mutation and recombination.
4. Update Phase: new agents replace the old population, and ecological parameters (e.g., prey abundance) are updated.
This explicit representation allows incorporation of spatial structure, stochasticity, and demographic fluctuations, which are difficult to model with purely deterministic equations. Moreover, ABM provides a natural setting for testing how system-level patterns --- such as emergent stooping behavior --- arise from micro-level rules.
3. Complementarity of Approaches
The replicator--mutator equation and ABM are not competing tools but complementary lenses. The former allows for mathematical analysis of equilibria, stability, and bifurcations. The latter enables simulation of emergent properties under realistic constraints. Together, they embody the CAS principle that evolution is both analyzable in aggregate and irreducible to averages, requiring hybrid approaches for full understanding.
D. Coupling with Predator--Prey Lotka--Volterra Extensions
Evolution does not occur in isolation but within the shifting ecological matrix of predator--prey interactions. To capture this, the genotype--phenotype--fitness mapping and replicator--mutator dynamics must be embedded in a broader ecological model. The natural starting point is the Lotka--Volterra framework, extended to incorporate trait dependency and evolutionary feedback.
1. Trait-Dependent Predation
Let NP(t)N_P(t)NP(t) and Nprey(t)N_{prey}(t)Nprey(t) denote predator and prey population sizes, respectively. Predation occurs at a rate determined not only by encounter frequency but also by the relative trait values of predator and prey. We define a trait-dependent predation function:
