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We consider an integrodifferential reaction-diffusion system on a multidimensional spatial domain subject to homogeneous Neumann boundary conditions.This system,which finds application in population dynamics,is characterized by non local delay terms depending on both the temporal and the spatial variables.The time delay effects are represented by memory kernels which are not necessarily monotone decreasing (i.e.,they can be of "strong" type).We first show how to construct a (dissipative) dynamical system on a suitable phase space.Then we discuss the existence of the global attractor as well as of an exponential attractor.