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A non-conserving zero-range process with extensive creation, annihilation andhopping rates is subjected to local resetting. The model is formulated on alarge, fully-connected network of states. The states are equipped with a(bounded) fitness level: particles are added to each state at a rateproportional to the fitness level of the state. Moreover, particles areannihilated at a constant rate, and hop at a fixed rate to a uniformly-drawnstate in the network. This model has been interpreted in terms of populationdynamics: the fitness is the reproductive fitness in a haploid population, andthe hopping process models mutation. It has also been interpreted as a model ofnetwork growth with a fixed set of nodes (in which particles occupying a stateare interpreted as links pointing to this state). In the absence of resetting,the model is known to reach a steady state, which in a certain limit mayexhibit a condensate at maximum fitness. If the model is subjected to globalresetting by annihilating all particles at Poisson-distributed times, there isno condensation in the steady state. If the system is subjected to localresetting, the occupation numbers of each state are reset to zero atindependent random times. These times are distributed according to a Poissonprocess whose rate (the resetting rate) depends on the fitness. We derive theevolution equation satisfied by the probability law of the occupation numbers.We calculate the average occupation numbers in the steady state. The existenceof a condensate is found to depend on the local behavior of the resetting rateat maximum fitness: if the resetting rate vanishes at least linearly at highfitness, a condensate appears at maximum fitness in the limit where the sum ofthe annihilation and hopping rates is equal to the maximum fitness.

Local resetting in non-conserving zero-range processes with extensive rates