Rising interest in the resilience of ecological systems has spawned diverse interpretations of the term's precise meaning, particularly in the context of resilience quantification. The purpose of this paper is twofold. The first aim is to use the language of dynamical systems to organize and scrutinize existing resilience definitions within a unified framework. The second aim is to provide an introduction for mathematicians to the ecological concept of resilience, a potential area for expanded quantitative research. To frame the discussion of resilience in dynamical systems terms, a model consisting of ordinary differential equations is assumed to represent the ecological system. The question “resilience of what to what?” posed by Carpenter et al. [2001] informs two broad categories of definitions, based on resilience to state variable perturbations and to parameter changes, respectively. Definitions of resilience to state variable perturbations include measures of basin size (relevant to one-time perturbations) and basin steepness (relevant to repeated perturbations). Resilience to parameter changes has been quantified by viewing parameters as state variables but has also considered the reversibility of parameter shifts. Quantifying this reversibility and fully describing how recovery rates determine resilience to repeated state-space perturbations emerge as two opportunities for mathematics research.

Ecosystem externalities arise when one use of an ecosystem affects its other uses through the production functions of the ecosystem. We use simulations with a size-spectrum ecosystem model to investigate the ecosystem externality created by fishing of multiple species. The model is based upon general ecological principles and is calibrated to the North Sea. Two fleets are considered: a “forage fish” fleet targeting species that mature at small sizes and a “large fish” fleet targeting large piscivorous species. Based on the marginal analysis of the present value of the rent, we develop a benefit indicator that explicitly divides the consequences of fishing into internal and external benefits. This analysis demonstrates that the forage fish fleet has a notable economic impact on the large fish fleet, but the reverse is not true. The impact can be either negative or positive, which entails that for optimal economic exploitation, the forage fishery has to be adjusted according to the large fish fishery. With the present large fish fishery in the North Sea, the two fisheries are well adjusted; however, the present combined exploitation level is too high to achieve optimal economic rents.

We applied a management strategy evaluation (MSE) model to examine the potential cost-effectiveness of using pheromone-baited trapping along with conventional lampricide treatment to manage invasive sea lamprey. Four pheromone-baited trapping strategies were modeled: (1) stream activation wherein pheromone was applied to existing traps to achieve 10^{−12} mol/L in-stream concentration, (2) stream activation plus two additional traps downstream with pheromone applied at 2.5 mg/hr (reverse-intercept approach), (3) trap activation wherein pheromone was applied at 10 mg/hr to existing traps, and (4) trap activation and reverse-intercept approach. Each new strategy was applied, with remaining funds applied to conventional lampricide control. Simulating deployment of these hybrid strategies on fourteen Lake Michigan streams resulted in increases of 17 and 11% (strategies 1 and 2) and decreases of 4 and 7% (strategies 3 and 4) of the lakewide mean abundance of adult sea lamprey relative to status quo. MSE revealed performance targets for trap efficacy to guide additional research because results indicate that combining lampricides and high efficacy trapping technologies can reduce sea lamprey abundance on average without increasing control costs.

Waterborne diseases are among the major health problems facing the world today. This is especially true in developing countries where there is limited access to clean water. In such settings, even when multiple water sources exist, they tend to be contaminated. In this paper, we formulate a waterborne disease model where individuals are exposed to multiple contaminated water sources. The fundamental mathematical features of the model such as the basic reproduction number and final epidemic size are obtained and analyzed accordingly. The global stability analysis of the disease-free equilibrium is performed. The model is later extended by considering vaccination as a possible control intervention strategy. An optimal control problem is constructed to investigate the existence of an optimal control function that reduces the spread of the disease with minimum cost. We support our analytical predictions by carrying out numerical simulations using published and estimated data from the recent cholera outbreak in Haiti.

In this paper, we study international river pollution problems. We introduce a model in which countries located along a river from upstream to downstream derive benefits from causing pollution, but also incur environmental costs from experiencing its own pollution and the pollution of all its upstream countries. The total welfare, being the sum of all benefits minus the sum of all costs, is maximized when all countries cooperate. Several principles from international water law are applied to find reasonable and fair distributions of the total welfare that can be obtained under full cooperation. Such a distribution of the welfare at efficient pollution levels can be implemented by monetary compensations.

Land transformation from grassland to cropland in the Northern Great Plains (NGP) has become a growing concern among many stakeholders. A growing body of work has sought to determine the amount and rate of land use change with less emphasis on the systemic structures or feedback processes of land use decisions. This paper presents the development of a system dynamics simulation model to integrate ecological, economic, and social components influencing land use decisions, including cattle ranching, cropland production, rural communities, land quality, and public policies. Evaluation indicated that the model satisfactorily predicted historical land, agricultural commodity, and rural community data from the model structure. Reference modes for key variables, including the farmland area, were characterized by a bias correction of 0.999, root mean squared error of prediction of 0.053, *R*^{2} of 0.921, and concordance correlation coefficient of 0.0959. The model was robust under extreme and varying sensitivity tests, as well as adequately predicting land use under changing system context. The model's major contributions were the inclusion of decision-making feedbacks from economic and social signals with connectivity to land quality and elasticity values that drive land transformation. Limitations include lack of spatial input and output capabilities useful for visual interfacing.

We consider an infinite time horizon spatially distributed optimal harvesting problem for a vegetation and soil water reaction diffusion system, with rainfall as the main external parameter. By Pontryagin's maximum principle, we derive the associated four-component canonical system (CS), and numerically analyze this and hence the optimal control problem in two steps. First, we numerically compute a rather rich bifurcation structure of flat (spatially homogeneous) canonical steady states and *patterned* canonical steady states (FCSS and PCSS, respectively), in 1D and 2D. Then, we compute time-dependent solutions of the CS that connect to some FCSS or PCSS. The method is efficient in dealing with nonunique canonical steady states, and thus also with multiple local maxima of the objective function. It turns out that over wide parameter regimes the FCSS, i.e., spatially uniform harvesting, are not optimal. Instead, controlling the system to a PCSS yields a higher profit. Moreover, compared to (a simple model of) private optimization, the social control gives a higher yield, and vegetation survives for much lower rainfall. In addition, the computation of the optimal (social) control gives an optimal tax to incorporate into the private optimization.

The waterborne diseases cause millions of deaths across the globe. It was a preconceived notion since years that ingestion of contaminated water is the only possible way for the spread of waterborne infectious diseases. But some recent studies have shown that waterborne disease can also spread as a result of human to human transmission. The use of disinfectants is a common practice to prevent a waterborne disease. We assume that the inclusion of the disinfectant, although helpful in prevention of disease, caused negative effect on individuals. In this paper, a nonlinear mathematical model has been proposed to analyze the negative effects caused by disinfectant of water on individuals. Our study shows that if the mixing of disinfectant has not been performed in a controlled manner, then it results in an increase in human to human transmission of disease. The equilibrium and stability analysis have been performed to study the nature of the model system. An extensive numerical experiment has been performed to support the analytical findings.

We compute the effects on the Alaska economy of reduced pollock harvests from rising sea surface temperature using a regional dynamic computable general equilibrium model coupled with a stochastic stock-yield projection model for eastern Bering Sea walleye pollock. We show that the effects of decreased pollock harvest are offset to some extent by increased pollock price, and that fuel costs and the world demand for the fish, as well as the reduced supply of the fish from rising sea surface temperature, are also important factors that determine the economic and welfare effects.