On existence, multiplicity, uniqueness and stability of positive solutions of a Leslie–Gower type diffusive predator–prey system

where u and v, respectively, represent the populations of the prey and the predator, and r, s, K, h are positive constants. The prey grows logistically with carrying capacity K and intrinsic growth rate r in the absence of predation. The predator consumes the prey according to the functional response f(u) and grows logistically with intrinsic growth rate s. The carrying capacity of the predator is proportional to the population size of the prey. The term hv/u is called the Leslie–Gower term. It measures the loss in the predator population due to rarity of its favorite food (see [2] for details). Problem (1) was studied extensively in recent years (see [2] for f is of Holling type I, see [3–6] for f is of Holling type II, see [7] for f is of Holling type III).