Spatial Mathematical Modeling (SoSe2026) — Interactive Simulations

Eleven interactive simulations of classic models in spatial ecology and evolutionary game theory, all running locally in your browser — no installation needed. Pick a demo below.

On-lattice logistic model (1D)

The classic birth-death interacting particle system on a ring of sites.

Classic interacting-particle-system model

Off-lattice logistic model

A birth-death process with density-dependent competition among individuals in continuous space. Choose between Gaussian and top-hat competition kernels and watch clustering vs. regularity emerge.

Surendran, Pinto-Ramos, Menezes & Martinez-Garcia, Physica D 477 (2025)

Colicin allelopathy (non-spatial)

The mean-field limit of the colicin lattice model: a phase portrait of producer vs. sensitive frequencies. Drag the cost and toxicity sliders to flip between monostable dominance and bistability, and click anywhere to launch a new trajectory.

Iwasa, Nakamaru & Levin, Evolutionary Ecology 12 (1998)

Colicin allelopathy (spatial)

Producer vs. sensitive cells on a lattice. Producers poison their neighbors — watch clumps of producers take over, or lose, depending on toxicity.

Durrett & Levin, J. Theor. Biol. 185 (1997)

Local vs. global dispersal

The E. coli C-S-R rock-paper-scissors game. Flip between local (Moore neighborhood) and well-mixed interaction and see biodiversity collapse under global dispersal.

Kerr, Riley, Feldman & Bohannan, Nature 418 (2002)

Mobility & biodiversity

Spatial rock-paper-scissors with selection, reproduction, and mobility (exchange). Cross the critical mobility and watch spiral waves collapse into a single surviving species.

Reichenbach, Mobilia & Frey, Nature 448 (2007)

Random walk → diffusion equation

Watch discrete random walkers fan out into a Gaussian, then compare the same walk against a finite-difference solution of the drift-diffusion PDE it converges to.

Classic random-walk / diffusion-limit demo

Habitat selection via movement

Beetles moving along a line of patches of varying quality. Compare a quality-aware Markov chain against a passive-diffusion null model and see habitat selection emerge from movement rates alone.

Kareiva, Ecological Monographs 52 (1982)

Step-selection functions

Simulate an animal's movement through habitat, then try to recover its hidden habitat-preference weights from the simulated steps.

Step-Selection Functions for Modeling Animal Movement

Semiarid vegetation patterning

A reaction-diffusion-advection model of water and plant biomass on a hillslope. Tune rainfall, advection, and diffusion to grow banded or spotted vegetation patterns, tracked live on a bifurcation diagram.

Klausmeier, Science 284 (1999)

Off-lattice spatial logistic model

Exact continuous-time spatial birth-death individual-based model (Gillespie SSA), periodic domain. Surendran, Pinto-Ramos, Menezes & Martinez-Garcia, Physica D 477 (2025), Section 2.1.
Plants: 0

Total population vs. time

On-lattice logistic model

Event-driven birth-death process on a ring of N sites.

Occupied fraction ρ(t) vs. sweep

Space-time raster (newest at top)

Colicin allelopathy lattice model

RED = colicin producer   BLUE = sensitive (non-producer)   white = vacant. Durrett & Levin, J. theor. Biol. 185 (1997).

Densities vs. time

Mobility & biodiversity (rock-paper-scissors)

RED = A   BLUE = B   GOLD = C   (A beats B beats C beats A)   black = vacant. Reichenbach, Mobilia & Frey, Nature 448 (2007).

Species densities vs. time

Local vs. global dispersal (C-S-R model)

RED = C (producer)   BLUE = S (sensitive)   GREEN = R (resistant)   white = vacant. Kerr, Riley, Feldman & Bohannan, Nature 418 (2002).

Strain densities vs. time

Colicin allelopathy — non-spatial (mean-field) model

Phase portrait of the complete-mixing limit of the colicin lattice model. Iwasa, Nakamaru & Levin, Evolutionary Ecology 12 (1998), Eq. 4.

Phase portrait (ρ1 vs. ρ2)

stable node unstable node saddle

Population dynamics for the trajectories at left

ρ1 (C, producer) ρ2 (S, sensitive)

Semiarid vegetation patterning

Water-biomass reaction-diffusion-advection model on a periodic hillslope. Klausmeier, "Regular and Irregular Patterns in Semiarid Vegetation", Science 284 (1999).

Plant biomass N(X,Y,t)

Bifurcation diagram (non-spatial model)

vegetated, stable vegetated, unstable bare soil, stable 2D sim. (mean N)

Random walk → diffusion equation

A discrete random walk, and the drift-diffusion PDE it converges to in the continuum limit.

Sample trajectories

Histogram of positions at this step

Random walk histogram vs. diffusion PDE

random walk histogram diffusion PDE (finite-difference) closed-form Gaussian

Habitat selection via movement

A linear array of patches; movement rate depends on local patch quality. Kareiva, "Experimental and Mathematical Analyses of Herbivore Movement", Ecological Monographs 52 (1982).

Step-selection function simulator

Embedded unmodified — identical to the standalone version.