This paper is concerned with the Cauchy problem for a system of parabolic equations which is derived from a complex-valued equation with a quadratic nonlinearity. First we show that if the convex hull of the image of initial data does not intersect the positive real axis, then the solution exists globally in time and converges to the trivial steady state. Next, on the onedimensional space, we provide some solutions with nontrivial imaginary parts that blow up simultaneously. Finally, we consider the case of asymptotically constant initial data and show that, depending on the limit, the solution blows up nonsimultaneously at space infinity or exists globally in time and converges to the trivial steady state.

This paper investigates the large time behaviour of a three species reaction-diffusion system, modelling the spatial invasion of two predators feeding on a single prey species. In addition to the competition for food, the two predators exhibit competitive interactions and, under some parameter condition, they can also be considered as two mutants. When mutations occur in the predator populations, the spatial spread of invasion takes place at a definite speed, identical for both mutants. When the two predators are not coupled through mutation, the spreading behaviour exhibits a more complex propagating pattern, including multiple layers with different speeds. In addition, some parameter conditions reveal situations where a nonlocal pulling phenomenon occurs and in particular where the spreading speed is not linearly determined.

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