Lattice Lotka–Volterra Models and Negative Cross-diffusion

Negative cross-diffusion has been identified as a factor which increases the possibility of spatio-temporal instabilities in Lotka-Volterra equations. But such terms have been considered quite rare in ecological modeling, since they imply deceitful prey-predator relationships. We show that negative cross-diffusion appears naturally in reaction diffusion equations obtained using a simple mean field decoupling technic on lattice Lotka-Volterra models. However, a linear stability analysis shows that spatial instabilities do not arise in any of the three models studied here. Two conditions leading to negative cross-diffusion and a possible reason for the absence of instabilities are also mentioned.