Katul, Gabriel G. ENV264 Applied Differential Equations in Environmental Sciences. Notes.
Many environmental problems require the construction and solution to equations (or systems of equations) that involve relations between rates and states of environmental variables. The objective of this course is to illustrate the use of differential equations and analytical tools to solve such problems as well as to interpret these solutions in the context of dynamical systems. The course covers basic analytical and numerical solutions to ordinary differential equations (ODE) with an introduction to partial differential equations commonly encountered in environmental studies (mainly diffusion and reaction-diffusion equations). Example applications include atomic waste disposal in oceans, refined population forecasting, hydrologic transport problems in xylems, predator-prey systems, heat transport in soils, and spatial models of biomass-water interactions. It is envisaged that this course will serve as an applied mathematics review for students who have not been in contact with calculus in the last 2-3 years. This course offers unique opportunities for professional and graduate students to be exposed to differential equations in a less ‘formal’ setting. Emphasis will be placed on a dynamical systems interpretation of differential equations via six case studies. Concepts such as stability, resilience, and equilibrium are routinely used in environmental sciences (climate, ecosystem, conservation efforts, etc...), yet the 'genesis' of these concepts remains imbedded in differential equations. Finally, ENV264 will make use of a number of computer-aided software (MATHEMATICA and Matlab) capable of solving analytically and numerically much of the ODEs encountered in environmental systems.
There follow six case studies with references which would serve as rich Modeling Scenarios.
Case study 1: Finite-time singularity in dynamics with application to world population,
economic and financial indices.
Johanson, A., and D. Sornette, 2001, Finite-time singularity in the dynamics of the
world population, economic and financial indices, Physica A, 294, 465-502.
Case study 2: Insect outbreak and introduction to Catastrophe Science.
Ludwig, D., D. Jones, and C.S. Holling, 1978, Qualitative analysis of insect
outbreak systems: The Spruce Budworm and Forest, Journal of Animal Ecology, 47, 315-332.
Case study 3: Catastrophic Shifts and Ecosystem Management.
Rietkerk et al., 2004, Self-Organized Patchiness and Catastrophic Shifts in
Ecosystems, Science, 305, 1926-1929.
Scheffer et al., 2001, Catastrophic Shifts in Ecosystems, Nature, 413, 591-596
Case study 4: The simple economics of Easter Island: A Ricardo-Malthus model of
renewable resource use.
Brander, J.A. and M.S. Taylor, 1998, The Simple Economics of Easter Island: A
Ricardo-Malthus Model of Renewable Resource Use, the American Economic Review, 88, 119-138.
Case study 5: Dynamics of resource production and utilisation in two-component
biosphere-human and terrestrial carbon systems
Raupach, M.R., 2007, Dynamics of resource production and utilisation in twocomponent
biosphere-human and terrestrial carbon systems, Hydrol. Earth Syst. Sci., 11, 875–889.
Case study 6: Dynamics of Love Affairs
Sprott, J.C., 2004, Dynamical models of love, Nonlinear Dynamics, Psychology, and Life Sciences, 8, 303-313.
Key words: differential equations, models, love, resource, compartment, economics, Easter island, insect, economics
Cite this work
Researchers should cite this work as follows: