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We consider a non-conserving zero-range process with hopping rateproportional to the number of particles at each site. Particles are added tothe system with a site-dependent creation rate, and removed from the systemwith a uniform annihilation rate. On a fully-connected lattice with a largenumber of sites, the mean-field geometry leads to a negative binomial law forthe number of particles at each site, with parameters depending on the hopping,creation and annihilation rates. This model of particles is mapped to a modelof population dynamics: the site label is interpreted as a level of fitness,the site-dependent creation rate is interpreted as a selection function, andthe hopping process is interpreted as the introduction of mutants. In the limitof large density, the fraction of the total population occupying each siteapproaches the limiting distribution in the house-of-cards model ofselection-mutation, introduced by Kingman. A single site can be occupied by amacroscopic fraction of the particles if the mutation rate is below a criticalvalue (which matches the critical value worked out in the house-of-cardsmodel). This feature generalises to classes of selection functions thatincrease sufficiently fast at high fitness. The process can be mapped to amodel of evolving networks, inspired by the Bianconi--Barab\'asi model, butinvolving a large and fixed set of nodes. Each node forms links at a ratebiased by its fitness, moreover links are destroyed at a uniform rate, andredirected at a certain rate. If this redirection rate matches the mutationrate, the number of links pointing to nodes of a given fitness level isdistributed as the numbers of particles in the non-conserving zero-rangeprocess. There is a finite critical redirection rate if the density of quenchedfitnesses goes to zero sufficiently fast at high fitness.

Steady states in a non-conserving zero-range process with extensive rates as a model for the balance of selection and mutation