Taking white noise, Lévy jumps, and periodic factors in the environments into account, we formulated a periodic stochastic Gompertz model with impulsive harvesting and investigated its optimal impulsive exploitation problem. For periodic stochastic ecosystems with Lévy jumps and impulsive harvesting, two traditional approaches, the Fokker-Planck equation approach and ergodicity-based approach used to study the optimal exploitation problems were invalid because for almost all such ecosystems, one could not solve the associated Fokker-Planck equations. In addition, the ecosystems have no traditional non-boundary invariant measures. The present study utilized a different method. By exploring the associated Hamilton function, we derived an optimal exploitation strategy. An example was also introduced to illustrate the theoretical results.

For the biased random walk on $d$-regular trees, we obtain the spectral radius, first return probability and $n$-step transition probability, speed. We prove that the spectral radius depends continuously on the bias parameter and is strictly increasing with respect to it, while the speed remains strictly positive and decreases monotonically as the bias increases.