The 3D version tracks a single well-mixed population number. This 2D companion instead spreads the population across a real landscape: the x-axis of the field is a temperature gradient (cold left, hot right), the y-axis is a rainfall gradient (wet top, dry bottom), and every point combines Shelford's bell-shaped tolerance curves for both factors with a fixed global soil-nutrient level via Liebig's law of the minimum to get a local fitness value.
response(x) = exp( −((x − optimum)² / (2·breadth²)) )
fitness(x,y) = min( response(temp(x)), response(water(y)), response(nutrient) )
Population density n(x,y,t) then follows a real Fisher-KPP reaction-diffusion equation — logistic growth toward the local fitness-scaled capacity, plus dispersal that lets organisms colonize neighboring cells:
∂n/∂t = D·∇²n + r_max·n·(fitness − n) [K_max = 1, no-flux boundaries]
That growth term is an exact algebraic simplification of the 3D model's r·fitness·N·(1−N/K) with K=K_max·fitness: the "fitness ×" and the "÷K" cancel to fitness−n, which stays perfectly smooth even where fitness→0 (outside an organism's tolerance) instead of dividing by a vanishing K.
- Climate shift — adds a uniform offset to the temperature gradient; watch the green "thriving" band visibly migrate toward the cool edge as the field warms, exactly the range-shift ecologists track under climate change.
- Dispersal D — the diffusion coefficient: low values leave isolated, sharply-bounded pockets; high values let population smear out and blur the tolerance boundary.
- Dominant limiting factor — per Liebig, whichever of temperature/water/nutrient has the lowest response is tallied across the whole grid; the majority vote is shown here.
- Reseed population — restarts every cell at a low uniform density (sparse colonizers), so you can watch the range re-form from scratch under the current climate.
Real-world relevance: this is the same class of model (reaction-diffusion with a heterogeneous, tolerance-derived carrying capacity) used to predict species range shifts under warming, invasive-species spread, and where a crop's viable growing zone sits between a cold and a dry edge.