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This paper explores stochastic differential equation models to characterize the growth dynamics of a stock modeled by generalized logistic growth under a general harvesting function. The latter function adds diminishing marginal productivity to effort increases, while the former incorporates several well‐known growth functions as particular instances. In order to obtain optimal policies, a Crank–Nicolson discretization scheme is derived for the general models. We employ the optimal control theory to determine the optimal harvesting policy for a specific harvested stock and compare the expected net present value of the harvester's total profit earned across harvesting policies. The main conclusion is that models with a higher growth rate result in greater profit values. [ABSTRACT FROM AUTHOR]

In recent times, there has been a notable global phenomenon characterized by a double predicament arising from the concomitant rise in worldwide life expectancy and a significant decrease in birth rates. The emergence of this phenomenon has posed a significant challenge for governments worldwide. It not only poses a threat to the continued viability of state-funded welfare programs, such as social security, but also indicates a potential decline in the future workforce and tax revenue, including contributions to social benefits. Given the anticipated escalation of these issues in the forthcoming decades, it is crucial to comprehensively examine the extension of the human lifespan to evaluate the magnitude of this matter. Recent research has focused on utilizing stochastic differential equations as a helpful means of describing the dynamic nature of mortality rates, in order to tackle this intricate issue. The usage of these models proves to be superior to deterministic ones due to their capacity to incorporate stochastic variations within the environment. This enables individuals to gain a more comprehensive understanding of the inherent uncertainty associated with future forecasts. The most important aims of this study are to fit and compare stochastic differential equation models for mortality (the geometric Brownian motion and the stochastic Gompertz model), conducting separate analyses for each age group and sex, in order to generate forecasts of the central mortality rates in France up until the year 2030. Additionally, this study aims to compare the outcomes obtained from fitting these models to the central mortality rates in Portugal. The results obtained from this work are quite promising since both stochastic differential equation models manage to replicate the decreasing central mortality rate phenomenon and provide plausible forecasts for future time and for both populations. Moreover, we also deduce that the performances of the models differ when analyzing both populations under study due to the significant contrast between the mortality dynamics of the countries under study, a consequence of both external factors (such as the effect of historical events on Portuguese and French mortality) and internal factors (behavioral effect). [ABSTRACT FROM AUTHOR]

We can describe the size evolution of a harvested population in a randomly varying environment using stochastic differential equations. Previously, we have compared the profit performance of four harvesting policies: (i) optimal variable effort policy, based on variable effort; (ii) optimal penalized variable effort policies, penalized versions based on including an artificial running energy cost on the effort; (iii) stepwise policies, staircase versions where the harvesting effort is determined at the beginning of each year (or of each biennium) and kept constant throughout that year (or biennium); (iv) constant harvesting effort sustainable policy, based on constant effort. They have different properties, so it is also worth looking at combinations of such policies and studying the single and cross-effects of the amount of penalization, the absence or presence and type of steps, and the restraints on minimum and maximum allowed efforts. Using data based on a real harvested population and considering a logistic growth model, we perform such a comparison study of pure and mixed policies in terms of profit, applicability, and other relevant properties. We end up answering the question: is the optimal enemy of the good? [ABSTRACT FROM AUTHOR]

The main goal of this work is to assess and compare the influence of Allee effects in profit optimization of stochastically fluctuating harvested populations considering several harvesting policies. The policies to compare are based on constant, variable, and stepwise harvesting efforts. For application purposes, the population growth models considered are the logistic model and a logistic-like model with weak Allee effects. In recent work, we have shown that the optimal harvesting policy with variable effort is inapplicable whereas the optimal harvesting policy with constant effort is easily applicable and leads to population sustainability. However, the latter implies profit losses, comparing to the first one. So, we consider a stepwise policy which is applicable but shares some of the problems with the optimal policy based on variable effort. We also show that some of the disadvantages of the optimal policy are eliminated by considering a penalized profit with an artificial running energy cost on the effort. However, the applicability problems remain. Finally, in terms of optimal profit, we study the influence of Allee effects on all policies and check whether Allee effects should or should not be taken into account when designing harvesting policies. [ABSTRACT FROM AUTHOR]

We describe the growth dynamics of a stock using stochastic differential equations with a generalized logistic growth model which encompasses several well-known growth functions as special cases. For each model, we compute the optimal variable effort policy and compare the expected net present value of the total profit earned by the harvester among policies. In addition, we further extend the study to include parameters sensitivity, such as the costs and volatility, and present an explicitly Crank–Nicolson discretization scheme necessary to obtain optimal policies. [ABSTRACT FROM AUTHOR]

• Uses a SDE Gompertz model for fish population growth in a random environment. • The optimal variable effort policy has inapplicable wildly varying efforts. • Two alternative policies are studied: constant effort and penalized variable effort. • Real data comparisons show that the constant effort policy is almost as profitable. • This confirms our previous findings using other data and the logistic model. In a previous paper, we discussed the use of an optimal variable effort fishing policy versus an optimal sustainable constant effort fishing policy in terms of the expected accumulated discounted profit during a finite time interval. We concluded that there is only a slight reduction in profit when choosing the applicable optimal sustainable constant effort fishing policy instead of the optimal variable effort fishing policy, which leads to major disadvantages in practice. In this paper, we confirm these conclusions by considering a different model, the Gompertz model, and by using another realistic dataset of parameters and a more general profit structure. We also show that some of the disadvantages of the optimal variable effort fishing policy, namely those related to social objectives, are eliminated by considering a penalized profit with an artificial running energy cost on the effort. However, the applicability problems remain. We also show that the profit advantage of this optimal penalized variable effort policy over the optimal sustainable constant effort policy is even smaller than the already very small advantage of the non-penalized policy. This further reinforces the robustness of our previous conclusions that the optimal sustainable constant effort policy, which does not have the shortcomings of the optimal variable effort fishing policy, is only slightly less profitable than it. [ABSTRACT FROM AUTHOR]

Linear mixed models provide a general and versatile approach for analyzing data collected in experiments, suitable for modeling repeated, longitudinal, or clustered observations. Important results for estimation can be obtained when subclasses of these mixed models are considered, based on some characteristics of their algebraic structure. The class of the models with commutative orthogonal block structure, for which least squares estimators are the best linear unbiased estimators, is of great interest. In an approach based on the algebraic structure of the models, and availing ourselves of U matrices, we study the possibility of extending a balanced mixed model, which could lead to a model with commutative orthogonal block structure. [ABSTRACT FROM AUTHOR]

We describe the growth dynamics of a harvested fish population in a random environment using a stochastic differential equation logistic model, where the harvest term depends on a constant or a variable fishing effort. We consider revenues to be proportional to the yield and costs to be quadratic in terms of effort. We compare the optimal expected profit obtained with two types of harvesting policies, one based on variable effort, which is inapplicable, and the other based on a constant effort, which is applicable and sustainable. We answer two new questions: (a) What is the constant effort that optimizes the expected profit per unit time? (b) How do the two policies compare in terms of performance? We show that, in a realistic situation, there is only a slight reduction in profit when choosing the applicable constant effort policy instead of the inapplicable policy with variable effort. [ABSTRACT FROM AUTHOR]

Stochastic differential equations are used to model the dynamics of harvested populations in random environments. The main goal of this work is to compute, for a particular fish population under constant effort harvesting, the mean and standard deviation of first passage times by several lower and upper thresholds values. We apply logistic or logistic-like with Allee effects average growth dynamics. In addition, we present a method to obtain the probability density function of the first passage time by a threshold through the numerical inversion of its Laplace transform. [ABSTRACT FROM AUTHOR]

There are a large number of commercially exploited stocks lacking quantitative assessments and reliable estimates of stock status. Providing MSY-based advice for these data-limited stocks remains a challenge for fisheries science. For many data-limited stocks, catch length composition and/or survey biomass indices or catch-per-unit effort (cpue) are available. Information on life history traits may also be available or borrowed from similar species/stocks. In this work we present three harvest control rules (HCRs), driven by indicators derived from key monitoring data. These were tested through simulation using two exploitation scenarios (development and over-exploitation) applied to 50 stocks (pelagic, demersal, deep sea species and Nephrops ). We examine the performance of the HCRs to deliver catch-based advice that is risk adverse and drives stocks to MSY. The HCR with a biomass index-adjusted status quo catch, used to provide catch-based advice for several European data-limited stocks, showed the poorest performance, keeping the biomass at low or very low levels. The HCRs that adjust the status quo catch based on the variability of the biomass index time series was able to drive most of the stocks to MSY, showing low to moderate biological risk. The recovery of biomass required asymmetric confidence intervals for the biomass index and larger decreases in status quo catch than increases. The HCR based on length reference points as proxies for the F SQ / F MSY ratio was able to reverse the decreasing trend in biomass but with levels of catch below MSY. This HCR did not prevent some of the stocks declining when subject to over-exploitation. For data-limited stocks, the empirical HCRs tested in this work can provide the basis for catch advice. Nevertheless, applications to real life cases require simulation testing to be carried out to tune the HCRs. Our approach to simulation testing can be used for such analysis. [ABSTRACT FROM AUTHOR]