In generalized Lotka-Volterra prey-predator models, many studies have been lead to analyze the influence of an alternative food source for the predator, both when this source has a fixed density [1] or is an alternative dynamic prey ([2] and subsequent works). Here, we will lead such a study with fixed density preys for two Leslie-Gower like models, where the prey population determines the carrying capacity of the predator population [3], where N and P are the dynamic prey and predator population densities and A is a constant alternative food source. The optimally foraging predator population will have to make a choice by taking some q 2 [0;1] between foraging the dynamic prey (q = 1), the other one (q = 0) or to adopt an intermediate strategy, depending on the respective prey densities. Its objective will be to maximize its per capita growth rate Pdot/P. Predators adopt pure strategies except when N = alpha_A A/alpha_N (both food sources produce the same carrying capacity), where a Filippov definition of the solutions can give rise to sliding modes. The behavior depends on whether the positive equilibrium with q = 1 is (i) stable or (ii) unstable or (iii) does not exist in models (17) and (18). Case (i) results in the maximization of the predator density at equilibrium. In case (ii), the alternative food source stabilizes the system which was unstable without it; it keeps the oscillations under control and results in a limit cycle that goes through a sliding mode. In case (iii) however, the other food source cannot prevent the populations to grow unbounded. In conclusion, the predator makes the most of the presence of an alternative food source by obtaining a maximal value for its equilibrium from it (case (i)) or gaining some stability (case (ii)). The effect on the dynamic prey N is not as straightforward; in particular, we show that no apparent competition principle [4] needs to hold.