Darboux integrability and dynamics of the Basener-Ross population model

We deal with the Basener and Ross model for the evolution of human population in Easter island. We study the Darboux integrability of this model and characterize all its global dynamics in the Poincaré disc, obtaining 15 different topological phase portraits

The characteristic approach in determining first integrals of a predator-prey system.

Master of Science in Mathematics, Statistics and Computer Science. University of KwaZulu-Natal, Durban 2016.Predator-Prey systems are an intriguing symbiosis of living species that interplay during the fluctuations of birth, growth and death during any period. In the light of understanding the behavioural patterns of the species, models are constructed via differential equations. These differential equations can be solved through a variety of techniques. We focus on applying the characteristic method via the multiplier approach. The multiplier is applied to the differential equation. This leads to a first integral which can be used to obtain a solution for the system under certain initial conditions. We then look at the comparison of first integrals by using two different approaches for various biological models. The method of the Jacobi Last Multiplier is used to obtain a Lagrangian. The Lagrangian can be used via Noether’s Theorem to obtain a first integral for the system

Environmental change and ecosystem functioning drive transitions in social-ecological systems: A stylized modelling approach

Sustainable management of social-ecological systems requires an understanding of how anthropogenic climate- and land use change may disrupt interactions between human societies and the ecosystem processes they depend on. In this study, we expand an existing stylized social-ecological system model by explicitly considering how urbanizing societies may become less dependent on local ecosystem functioning. This expansion is motivated by a previously developed conceptual framework suggesting that societies may reside in either a green loop and be strongly dependent on local ecosystem processes, or in a red loop where this dependency is weaker due to imports of natural resources from elsewhere. Analyzing the feasibility and stability of local social-ecological system states over a wide range of environmental and socio-economic conditions, we observed dynamics consistent with the notion of green loop-dominated and red loop-dominated societies comprising alternate stable social-ecological states. Based on systems' inherent dependencies on local ecosystem processes, responses to environmental change could comprise either transitions between green loop- and red loop-dominated states, or collapse of either of these states. Our quantitative model provides an internally consistent mapping of green loop- and red loop-dominated states, as well as transitions between or collapses of these states, along a gradient of environmental conditions