(TP,tprey)=phunt(TP,tprey),\phi(\bar{\mathbf{T}}_P, \bar{\mathbf{t}}_{prey}) \;=\; \kappa \cdot p_{hunt}(\bar{\mathbf{T}}_P, \bar{\mathbf{t}}_{prey}),(TP,tprey)=phunt(TP,tprey),
where:
TP\bar{\mathbf{T}}_PTP is the mean predator phenotype vector,
tprey\bar{\mathbf{t}}_{prey}tprey is the mean prey phenotype vector,
phuntp_{hunt}phunt is the hunting success probability defined in Section III.B,
\kappa is a scaling constant converting probability into predation rate.
This ensures that ecological interactions are explicitly coupled to evolutionary traits.
2. Extended Lotka--Volterra Equations
The coupled population dynamics are:
dNpreydt=rpreyNprey(1NpreyK)(TP,tprey)NPNprey,\frac{dN_{prey}}{dt} = r_{prey} N_{prey} \left(1 - \frac{N_{prey}}{K}\right) - \phi(\bar{\mathbf{T}}_P, \bar{\mathbf{t}}_{prey}) \, N_P \, N_{prey},dtdNprey=rpreyNprey(1KNprey)(TP,tprey)NPNprey, dNPdt=e(TP,tprey)NPNpreydNP,\frac{dN_P}{dt} = e \, \phi(\bar{\mathbf{T}}_P, \bar{\mathbf{t}}_{prey}) \, N_P \, N_{prey} - d \, N_P,dtdNP=e(TP,tprey)NPNpreydNP,
where:
rpreyr_{prey}rprey is prey intrinsic growth rate,
KKK is carrying capacity,
eee is conversion efficiency of prey biomass into predator reproduction,
ddd is predator mortality rate.
These equations generalize classical Lotka--Volterra by embedding evolutionary state variables into the predation function.
3. Eco-Evolutionary Feedback
Crucially, the mean traits TP\bar{\mathbf{T}}_PTP and tprey\bar{\mathbf{t}}_{prey}tprey are not static but evolve through replicator--mutator dynamics (Section III.C). Thus, the ecological system feeds back into evolutionary trajectories:
If prey evolve increased speed or evasiveness, predator hunting probability decreases, reducing predator population size.
If predators evolve enhanced aerodynamic or sensory traits, prey mortality rises, feeding back to alter prey population structure.
Demographic bottlenecks in either species alter genetic drift and the fixation of adaptive alleles.
This eco-evolutionary coupling generates Red Queen dynamics: continuous adaptation without permanent resolution. It also provides a natural explanation for punctuated shifts: once ecological thresholds are crossed, trait distributions can reorganize rapidly, producing bursts of evolutionary change.
