2. Attractor Dynamics
Mathematically, these emergent attractors appear as stable equilibria or limit cycles in the coupled replicator--mutator and Lotka--Volterra system. Populations initially disperse across genotype and phenotype space, but feedback from selection and ecological constraints narrows variation into phenotypic clusters. Depending on parameter regimes, the system may settle into:
Stable attractors, representing sustained coordinated adaptations.
Oscillatory attractors, reflecting predator--prey Red Queen dynamics where predator and prey traits coevolve in cycles.
Multistability, where different coordinated configurations coexist and populations may stochastically transition between them.
3. Punctuated Emergence
Simulations frequently revealed periods of slow, incremental change punctuated by sudden reorganizations of trait distributions, consistent with punctuated equilibrium. These shifts often occurred when rare mutational combinations unlocked previously inaccessible trait synergies, allowing populations to escape local adaptive valleys and converge rapidly on higher-fitness attractors. Bottleneck scenarios amplified this effect, as reduced diversity facilitated the fixation of coordinated allele sets.
4. Biological Interpretation
The emergent attractor corresponding to the peregrine falcon's phenotype illustrates how CAS dynamics resolve the adaptive puzzle of synchronization. Rather than requiring that each trait evolve independently with immediate benefit, the CAS framework shows that self-organization of genetic networks under ecological feedback can drive traits to align into functional modules. The falcon's aerodynamic, sensory, physiological, and neuromuscular systems thus represent not an improbable coincidence of parallel adaptations, but the natural outcome of attractor dynamics in a complex adaptive system.
V. Results
A. Analytical Results: Stability, Bifurcations, Attractors
The coupled replicator--mutator and trait-dependent Lotka--Volterra system exhibits rich nonlinear dynamics characteristic of complex adaptive systems. Analytical exploration of reduced forms and stability conditions revealed several key behaviors that provide theoretical grounding for the simulation results.
1. Stability of Coordinated Trait Configurations
Equilibria of the system were identified by solving for stationary distributions of genotype frequencies and trait means under constant ecological conditions. Stability analysis, via Jacobian eigenvalue evaluation around equilibria, showed that isolated improvements in single traits often produced unstable fixed points. Only when multiple traits simultaneously exceeded threshold values did stable equilibria emerge, corresponding to coordinated adaptations. This supports the attractor hypothesis: fitness landscapes contain basins of attraction defined by multi-trait synergies, not by isolated trait optima.
