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Develop a PDE-based model that describes the dynamics of a predator-prey ecosystem with the following assumptions: • the prey population grows following the logistic growth model

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Develop a PDE-based model that describes the dynamics of a predator-prey ecosystem with the following assumptions:

• the prey population grows following the logistic growth model.

• the predator population tends to decay naturally without prey.

• predation (i.e., encounter of prey and predator populations) increases the predator population while it decreases the prey population.

• the predator population tends to move toward areas with high prey density.

• the prey population tends to move away from areas with high predator density.

• both populations diffuse over space.