MkM_{k\ell}Mk represents the pleiotropic matrix, quantifying the effect of locus kkk on trait \ell. A single locus may therefore contribute to multiple traits.
EkjE_{kj\ell}Ekj encodes epistatic coefficients, capturing how the joint state of loci kkk and jjj influences trait \ell. These terms introduce nonlinearity and context-dependence into trait formation.
\eta_\ell represents stochastic influences such as developmental noise or environmental plasticity, modeled as a random variable (e.g., Gaussian with variance 2\sigma^22).
This formalism generalizes the standard additive model of quantitative genetics. If Ekj=0E_{kj\ell} = 0Ekj=0 and each locus maps to only one trait, the model reduces to the classical linear mapping. By contrast, when pleiotropy and epistasis are nonzero, traits become interdependent, and coordinated adaptive packages can emerge or collapse depending on the joint configuration of alleles.
Such a mapping naturally generates rugged fitness landscapes: different genotypic combinations may yield similar trait values, multiple genotypes may map to high-fitness phenotypes, and small genetic changes can result in large phenotypic effects. These properties are hallmarks of CAS, where system-level behavior arises not from single components but from structured interactions.
To illustrate, consider a simplified model with four key traits for high-speed predation:
T1T_1T1: visual acuity,
T2T_2T2: respiratory capacity,
T3T_3T3: neuromuscular control,
T4T_4T4: aerodynamic wing structure.
Each is influenced by overlapping subsets of loci. For example, a gene involved in oxygen transport may directly affect T2T_2T2 but also indirectly constrain T3T_3T3 by modulating metabolic output. Another gene regulating feather microstructure may enhance T4T_4T4 while incurring aerodynamic costs if not complemented by neuromuscular adaptation (T3T_3T3). Only when specific allele sets align do these traits jointly support the stooping behavior of the peregrine falcon.
This genotype--phenotype formalism establishes the foundation for the fitness mapping that follows, where traits are evaluated not in isolation but in their coordinated contribution to survival and reproduction under ecological pressures.
B. Fitness Function with Trade-Offs and Ecological Dependency
Having established a formal mapping from genotype to phenotype, the next step is to quantify how trait configurations translate into reproductive success. In evolutionary biology, fitness is not a static property of an individual but a context-dependent measure shaped by ecological interactions and by the energetic costs associated with maintaining adaptive traits.
We define the fitness of an individual with trait vector T\mathbf{T}T in ecological context eco\Theta_{eco}eco as:
w(T,eco)=exp(s[B(T,eco)C(T)]),w(\mathbf{T}, \Theta_{eco}) \;=\; \exp \Big( s \cdot \big[ B(\mathbf{T}, \Theta_{eco}) - C(\mathbf{T}) \big] \Big),w(T,eco)=exp(s[B(T,eco)C(T)]),
where:
B(T,eco)B(\mathbf{T}, \Theta_{eco})B(T,eco) represents the benefit function, capturing the probability of successful predation and energy intake under given ecological conditions.
C(T)C(\mathbf{T})C(T) denotes the cost function, accounting for the metabolic and structural trade-offs of maintaining particular traits.
sss is a scaling parameter representing the strength of selection.
The exponential form ensures non-negative fitness values and reflects multiplicative accumulation of selection across generations.
1. Benefit Function: Predatory Success
