Co-evolution in Prey–Predator Systems

• The framework rigorously integrates stochastic birth–death–mutation dynamics with deterministic ODEs to capture evolutionary trajectories in prey–predator systems.
• Trait-structured dynamics model key ecological interactions including competition, predation rates, and invasion fitness to predict shifts in community structure.
• Simulations demonstrate emergent phenomena such as arms races, evolutionary murder, and trait-mediated extinctions driven by feedback between ecological and evolutionary processes.

A co-evolutionary framework in Prey–Predator Systems (PBS) rigorously integrates individual-based stochastic birth–death–mutation processes with deterministic ecological and evolutionary theory to analyze phenotypic dynamics in interacting prey and predator populations. This approach provides a multiscale hierarchy linking microscopic stochasticity, population-level fluctuations, and macroscopic evolutionary trajectories, with explicit focus on trait-dependent ecological interactions, invasion fitness, and the emergence of equilibria under natural selection.

1. Model Specification and Trait-Structured Dynamics

The model is grounded in a continuous-time, individual-based stochastic process. Each prey carries a phenotypic trait $x$—potentially multidimensional, e.g., $x = (q_a, q_n)$ encoding qualitative ($q_a$) and quantitative ($q_n$) defense mechanisms—while each predator carries a trait $y$—commonly $y = (\rho, \sigma)$, with $\rho$ the prey-preference parameter and $\sigma$ denoting generalism.

Prey birth rate is $b(x)$, possibly reduced by fitness trade-offs (e.g., cost of defense); death rate is

$$\lambda(x, Z^K) = d(x) + \sum_{x'} c(x, x') n_{x'} + \sum_{y} B(x, y) h_y$$

where $d(x)$ is the intrinsic death rate, $c(x, x')$ the competition between prey of types $x$ and $x'$, and $B(x, y)$ the interaction strength (predation rate) between prey type $x$ and predator $y$. $n_{x'}$ and $h_y$ are normalized densities.

Predator birth is proportional to prey consumption: $r \sum_{i=1}^d \frac{B(x_i, y)}{K} N^K_{x_i}$ with death at rate $D(y)$.

The full community state is

$$Z^K = \frac{1}{K} (N^K_1, ..., N^K_d, H^K_1, ..., H^K_m)$$

where $K$ denotes the system size.

2. Eco-evolutionary Mutation and Invasion Dynamics

At each birth event, mutations occur with probability $u_K \ll 1$, shifting offspring phenotype according to a kernel (typically Gaussian increments). The crucial evolutionary notion is the mutant’s "invasion fitness," defined for prey: $$s(x'; z^*) = b(x') - d(x') - \sum_{j} c(x', x_j) n^*_j - \sum_{k} B(x', y_k) h^*_k$$ where $z^* = (n^*, h^*)$ is the equilibrium community.

If $s(x'; z^*) > 0$, the mutant can successfully invade, leading to a rapid stochastic transition ("jump") to a new equilibrium—a process captured mathematically by a Markovian jump process, generalizing the Polymorphic Evolutionary Sequence (PES) to full prey–predator co-evolution.

3. Deterministic Limits and ODE System

Under $K \to \infty$, with mutations neglected on demographic time scales, the model converges to a deterministic system of ODEs with $d$ prey and $m$ predator types:

• Prey: $$\displaystyle \frac{dn_i}{dt} = n_i \left[b(x_i) - d(x_i) - \sum_j c(x_i, x_j) n_j - \sum_l B(x_i, y_l) h_l \right]$$
• Predator: $$\displaystyle \frac{dh_l}{dt} = h_l \left[r \sum_i B(x_i, y_l) n_i - D(y_l)\right]$$

Rigorous convergence follows via Poisson point process representations and Lyapunov function techniques, connecting stochastic trajectories to deterministic mean field equations.

4. Equilibrium and Stability Analysis

Uniqueness and global stability of the coexistence equilibrium are established under conditions excluding "invasible" missing types:

• For any extinct prey $x_i$: $$b(x_i) - d(x_i) - \sum_j c(x_i, x_j)n_j^* - \sum_l B(x_i, y_l) h^*_l < 0$$
• For any extinct predator $y_l$: $r \sum_i B(x_i, y_l) n^*_i - D(y_l) < 0$

Stability is proven by constructing a Lyapunov function: $$V(z) = \sum_i r \left[n_i - n_i^* \log n_i\right] + \sum_l (h_l - h_l^* \log h_l)$$ with further quadratic corrections. The connection to solvability of the associated Linear Complementarity Problem (LCP) for the interaction matrix is central in guaranteeing both uniqueness and attractivity of equilibria.

5. Mutation Timescale: Jump Process and Adaptive Dynamics

In the rare mutation limit ($u_K \ll 1/K$), the long-term evolution is governed by a jump process: the community resides at an equilibrium until a successful mutant invades (with probability determined by its fitness advantage), after which a rapid transition (on timescale $O(\log K)$) establishes a new equilibrium.

When mutation step-sizes $\epsilon$ are also small, trait evolution follows a co-evolutionary canonical equation: $$\begin{align*} \frac{dx}{dt} &= \frac{1}{2} p(x) \gamma(x) n^*(x, y) \nabla_x s(x; (x, y)) \ \frac{dy}{dt} &= \frac{1}{2} P(y) \Gamma(y) h^*(x, y) \nabla_y F(y; (x, y)) \end{align*}$$ where $p(x)$, $P(y)$ are mutation rates, $\gamma(x)$, $\Gamma(y)$ covariance matrices, and $n^*(x, y)$, $h^*(x, y)$ are resident equilibrium densities. This system captures the co-adaptive dynamics of prey and predator trait means.

6. Simulation Results and Evolutionary Scenarios

Numerical experiments elucidate several nontrivial eco-evolutionary phenomena:

• Coevolution of prey qualitative defense $q_a$ and predator preference $\rho$ can result in directional selection against predation. In the absence of predator evolution, prey may evolve defense values that drive predators extinct ("evolutionary murder").
• Allowing both prey and predator mutation yields dynamic regimes ranging from arms-race (continuous reciprocal trait shifts) to convergence, dependent on mutation rates and variances.
• Evolution of quantitative defense $q_n$ is settlement-dependent on the associated reproduction trade-off; high defense cost can reversibly reduce defense investment, modulating predator persistence. The simulations robustly demonstrate that trait evolution may drastically alter trophic structure and interaction strength, including complete elimination of a trophic level.

7. Multiscale and Mechanistic Insights

The framework rigorously links three scales:

• Micro: stochastic individual-level dynamics, incorporating mutation and demographic noise,
• Meso: deterministic ecological equilibria emerging in large populations,
• Macro: trait evolution under rare or small mutations (Markovian jump process and canonical adaptive dynamics ODEs). This multiscale perspective exposes explicit mechanistic routes—via invasion fitness landscapes, trait–interaction feedbacks, and population dynamical stability—by which natural selection modulates trait distributions and community structure.

Conclusion

The co-evolutionary framework in PBS systematically formalizes the feedbacks between phenotypic trait evolution and population dynamics in interacting communities, extending classical Lotka–Volterra models to a full eco-evolutionary context. It elucidates how the trajectory and outcome of natural selection are shaped by interspecific interactions, defense costs, and the stochasticity of mutation-driven invasion. The hierarchy of deterministic and stochastic limits, global stability proofs, and simulation-based demonstration of emergent phenomena such as evolutionary murder, arms races, and trait-mediated extinctions establish this approach as a robust and predictive tool for the analysis of co-evolution in complex ecosystems (Costa et al., 2014).