[Paper Review] Invasion and adaptive evolution for individual-based spatially structured populations

Nicolas Champagnat, Sylvie Méléard|arXiv (Cornell University)|Jan 1, 2006

Mathematical and Theoretical Epidemiology and Ecology Models1 references8 citations

TL;DR

This paper develops a stochastic individual-based model for spatially structured populations with adaptive evolution, incorporating birth, death, mutation, competition, and spatial diffusion. It demonstrates that scaling the spatial interaction range leads to convergence from nonlocal to local reaction-diffusion equations, revealing how nonlocal interactions promote clustering and polymorphism, while local interactions accelerate invasion through trait evolution.

ABSTRACT

The interplay between space and evolution is an important issue in population dynamics, that is in particular crucial in the emergence of polymorphism and spatial patterns. Recently, biological studies suggest that invasion and evolution are closely related. Here we model the interplay between space and evolution starting with an individual-based approach and show the important role of parameter scalings on clustering and invasion. We consider a stochastic discrete model with birth, death, competition, mutation and spatial diffusion, where all the parameters may depend both on the position and on the trait of individuals. The spatial motion is driven by a reflected diffusion in a bounded domain. The interaction is modelled as a trait competition between individuals within a given spatial interaction range. First, we give an algorithmic construction of the process. Next, we obtain large population approximations, as weak solutions of nonlinear reaction-diffusion equations with Neumann's boundary conditions. As the spatial interaction range is fixed, the nonlinearity is nonlocal. Then, we make the interaction range decrease to zero and prove the convergence to spatially localized nonlinear reaction-diffusion equations, with Neumann's boundary conditions. Finally, simulations based on the microscopic individual-based model are given, illustrating the strong effects of the spatial interaction range on the emergence of spatial and phenotypic diversity (clustering and polymorphism) and on the interplay between invasion and evolution. The simulations focus on the qualitative differences between local and nonlocal interactions.

Motivation & Objective

• To model the interplay between spatial structure and adaptive evolution in asexual populations with individual-level dynamics.
• To investigate how spatial interaction range influences clustering, polymorphism, and invasion dynamics.
• To derive deterministic limits of individual-based models as population size increases, linking microscopic mechanisms to macroscopic PDEs.
• To analyze the transition from nonlocal to local interactions in reaction-diffusion equations via parameter scaling.
• To simulate and compare the qualitative effects of local versus nonlocal competition on trait and spatial diversification.

Proposed method

• Constructs a Markovian point process with individual-level events: birth, death, mutation, and movement via reflected diffusion.
• Models spatial movement using reflected diffusion in a bounded domain to enforce spatial confinement.
• Implements trait-dependent competition within a spatial interaction range δ, using a kernel Iδ(y) and competition weight W(v).
• Derives large population limits by scaling population size N and interaction range δ, leading to weak solutions of nonlinear reaction-diffusion equations.
• Establishes convergence to nonlocal PDEs when δ is fixed, and to local PDEs as δ → 0, with Neumann boundary conditions.
• Uses individual-based simulations to explore emergent patterns under varying interaction ranges and initial conditions.

Experimental results

Research questions

• RQ1 How does the spatial interaction range influence the emergence of spatial clustering and phenotypic polymorphism?
• RQ2 What is the relationship between adaptive evolution and invasion speed in spatially structured populations?
• RQ3 How do nonlocal and local competition mechanisms lead to different dynamical outcomes in trait and space evolution?
• RQ4 What are the limiting deterministic PDEs that describe the large-population behavior of the individual-based stochastic model?
• RQ5 How does the initial distribution of traits and positions affect the long-term invasion and diversification patterns?

Key findings

• Nonlocal interactions (fixed δ > 0) lead to the emergence of spatial and phenotypic clusters, with multiple clusters spreading across the trait space over time.
• As the interaction range δ decreases to zero, the system converges to a local reaction-diffusion equation with Neumann boundary conditions, indicating a transition from nonlocal to local competition effects.
• Simulations show that when initial individuals are slow-moving and low-traits, invasion spreads slowly in physical space but accelerates as fitter traits (e.g., higher dispersal speed) evolve and spread.
• With larger interaction ranges δ, two distinct clusters emerge rapidly in the trait space and spread spatially, forming branched structures linking the initial and extreme trait clusters.
• The invasion front is composed of faster-dispersing individuals, indicating that evolutionary adaptation of dispersal speed directly enhances spatial invasion speed.
• The model reveals that evolution and invasion are tightly coupled: the spread of new traits drives the expansion of the population front, especially when initial traits are suboptimal.