2. Bifurcations and Punctuated Dynamics
As parameters such as mutation rate (\mu), selection strength (sss), or prey evasiveness were varied, equilibria underwent bifurcations. Specifically:
Saddle-node bifurcations occurred when incremental improvements in trait synergy collapsed into coordinated attractors.
Hopf bifurcations produced oscillatory predator--prey dynamics, reflecting Red Queen cycles.
Pitchfork bifurcations were observed in multistable regimes, where populations could stabilize around distinct but functionally equivalent coordinated trait bundles.
These bifurcations explain the observed pattern of punctuated equilibrium: long periods of stability interrupted by sudden shifts as populations reorganize around new attractors.
3. Role of Epistasis and Pleiotropy
Analytical simplification of the genotype--phenotype mapping revealed that epistasis introduces cross-terms that can destabilize single-trait optima, while pleiotropy couples improvements across traits. Together, they create the mathematical conditions for synchronized adaptation. In the absence of these interactions, equilibria tended to be shallow and easily destabilized, leading to slow, uncoordinated change.
4. Eco-Evolutionary Coupling
Embedding the predator--prey system within the Lotka--Volterra extension produced feedback dynamics where prey adaptation shifted predator equilibria and vice versa. Stability analysis showed that:
High prey evasiveness raised the threshold for predator coordination, deepening adaptive valleys.
Predator trait improvements destabilized prey equilibria, often leading to cycles rather than fixed points.
Parameter regions of co-stability existed, where both predator and prey trait distributions stabilized, but only when trait costs enforced balanced trade-offs.
5. Emergent Attractors as Evolutionary Blueprints
The analysis demonstrated that attractors corresponding to the peregrine falcon's phenotype are not singular, improbable solutions, but recurrent outcomes across broad parameter ranges. Multiple attractor basins were identified, each representing alternative coordinated phenotypes. The peregrine-like attractor was characterized by simultaneous optimization of vision, respiration, neuromuscular control, and wing morphology, providing a theoretical foundation for its emergence and persistence in nature.
B. Simulation Results: Trajectories of Allele Frequencies, Trait Synchronization, Red Queen Cycles
1. Allele Frequency Dynamics
