Mathematical Models in Social Ecological Systems

This simple model was developed by Edward Lorenz in 1963 to study fluid mechanics (based on Navier-Stokes equations). It is the first ever model of a chaotic dynamical system. Chaos arises when a deterministic, nonlinear dynamical system exhibits long-term unpredictability in behavior due to sensitivity to initial conditions.

This model describes the dynamics of a waterwheel. The wheel has 10 cups that are filled when as they pass over the top of the arc of the wheel (imagine a Ferris wheel where instead of seats there are buckets. Water is flowing down on the Ferris wheel filling the buckets). The weight of the water makes the wheel turn. However, in this model, the buckets leak. So if the flow of the water from the top is too slow, some interesting motion can occur.

This is a simple predator-prey model with type I or II functional response (depending on parameter choices). The model is also known as the Lotka–Volterra equation. Prey grows logistically and is harvested by predators. In the model, predators are specialists (i.e., they eat only one particular prey species for survival and growth), and their predation pattern can be either type I or II functional response. In type I, the predation rate depends only on the prey density ($b=0$), i.e., how abundant the prey is.

The logistic is the simplest representation of population dynamics  that contains a minimum level of biological realism.  If we imagine a  population of individuals at time t, the number of individuals in the next period (periods can range from seconds to days to decades depending on the organism) is the number of individuals now plus the number of births minus the number of deaths.  If we assume in the simplest case there are no environmental constraints on births or deaths, then it is reasonable to assume that a constant proportion of living individuals give birth (i.

This is a simple model for resource dynamics under harvest. The authors added expected effects of investment in fixed structures on the dynamics of settlement. These sunk-cost effects are not included in this simple model.

This discrete time model represents a coupled system as seen through talk turns in a conversation between a wife and husband where each exerts influence dynamically on the other. Cook et al.

Exploring the dynamic feedbacks between biological and cultural evolutionary systems is critical to understanding the origins of modern human behavior. The authors present a population dynamics model using ordinary differential equations and agent based modeling to delineate the consequences of differing mobility strategies expressed as reproduction potential at the population level for Late Pleistocene hominins in Western Eurasia (i.e., modern Homo sapiens and Neanderthals).

In this case, the surface water level is modeled by a first order differential equation. There are six sections with equal area that are allocated water based on institutions that determine how long each sector receives water, and in what order. Water is allocated to a given sector by opening and closing gates along the irrigation canals. Based on these allocations, each sector maintains a crop yield depending on the amount of time that maintains their water level within the bounds of a desired water height.

The dynamic model for poverty traps contains parameters as government policies and technologies, and variables such as income, expenditures, assets holdings and anthropometric status. This dynamic model for poverty traps offers a new perspective by showing the existence of multiple stable equilibria, which implies that there's one unstable dynamic equilibria. At this unstable equilibria, any little shock in the system will cause a switch to state equilibrium.