The stochastic logistic model is widely used to capture random fluctuations arising from birth-death processes in ecological populations. We use this model to study the impact of environmental perturbations that may occur naturally or as a consequence of population harvesting. In our model formulation, environmental perturbations occur randomly as per a Poisson process, and the perturbations result in each individual dying with a certain probability of death. Moment closure schemes are employed to derive expressions for the mean and variability in population numbers. Moreover, to quantify the impact of population extinction in our model we compute the probability of extinction using the Finite State Projection (FSP) numerical method. Our analysis shows that rare environmental perturbations with a high probability of death lead to overall larger random fluctuations and extinction risk as compared to frequent perturbations with a low probability of death. Finally, we formulate the problem in the context of population harvesting to find the optimal harvesting rate that maximizes the cumulative yield.

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The timing of biochemical events is often determined by the accumulation of a protein or chemical species to a critical threshold level. In a stochastic model, we define event timing as the first-passage time for the level to cross the threshold from zero or random initial conditions. This first-passage time can be modulated by implementing feedback in synthesis, that is, making the production rate an arbitrary function of the current species level. We aim to find the optimal feedback strategy that reduces the timing noise around a given mean first-passage time. Previous results have shown that while a no-feedback strategy (i.e., an independent constant production rate) is optimal in the absence of degradation and zero-molecules initial condition, a negative feedback is optimal when the process starts at random initial conditions. We show that when the species can be degraded and the synthesis rates are set to depend linearly on the number of molecules, a positive feedback strategy (the production rate increases with the level of the molecule) minimizes timing noise. However, if no constraints on the feedback are imposed, the optimal strategy involves a mixed feedback approach, which consists of an initial positive feedback followed by a sharp negative feedback (the production rate decreases with the level) near the threshold. Finally, we quantify the fundamental limits of timing noise reduction with and without feedback control when time-keeping species are subject to degradation.

What is the optimal way to harvest an ecological population sustainably is a fundamental problem in natural resource management. Here we use the framework of the stochastic logistic model which captures random birth-death of individuals to determine the optimal harvesting strategy that maximizes the integrated yield over time. Harvesting is assumed to occur at either a constant or state-dependent rate, and individuals are harvested with a certain probability whenever a harvesting event occurs. A special case of state-dependent harvesting is a threshold-based strategy, where harvesting is done when the population crosses a threshold. We use moment closure schemes to develop analytical formulas quantifying the mean and optimal yield. Moreover, as populations are susceptible to extinction at high harvesting rates, the Finite State Projection (FSP) method is used to estimate the probabilities of extinction across strategies and parameter regimes. Our results show that the threshold-based strategy is most effective in maximizing the yield as it suppresses population fluctuations and minimizes extinction events.

We consider the problem of transmitting the state value of a dynamical system through a communication network. The dynamics of the error in state estimation is modeled using a stochastic hybrid system formalism, where the error grows exponentially over time. Transmission occurs over the network at specific times to acquire the system’s state, and whenever a transmission is triggered, the error is reset to a zero-mean random variable. Our goal is to uncover transmission strategies that minimize a combination of the steady-state error variance and the average number of transmissions per unit of time. We find that a constant Poisson rate of transmission results in a heavy-tailed distribution for the estimation error. Next, we consider a random non-threshold transmission rate that varies as a power law of the error. Finally, we explore a threshold- based rate in which transmission occurs exactly when the error reaches a threshold. Our results show that if the error’s variance after transmission is small enough, a threshold-based strategy is the optimal paradigm. On the other hand, if this variance is large, and the error does not grow fast enough, the random non-threshold transmission strategy emerges as optimal. These analytical results are verified by simulations of the stochastic hybrid system.