Abstract
In this chapter, we use ecological scenarios as settings in which to develop and study models for the change of populations over time. We restrict ourselves for now to models that require careful monitoring of only one population. There are two main categories of dynamic models, discrete and continuous, differing in the assumption made about how to mark time. Discrete dynamic models assume that time can be broken up into distinct uniform intervals. The length of the interval depends on the life history of the organism being modeled. Salmon have yearly spawning periods, so a time interval of 1 year is chosen for a discrete salmon model. Continuous dynamic models assume that time flows continuously from one moment to the next. The assumption of continuity in time is relative to the overall duration of the population. For example, it is common to ignore the diurnal variation of temperature and sunlight in a model that tracks a population of plants over a complete growing season.
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Notes
- 1.
- 2.
“Recruitment” is a term biologists use to describe the combined processes of birth and survival to the next census.
- 3.
For an amusing illustration of this principle, see the short story Pigs is Pigs [2].
- 4.
This is always true if there is no migration from outside.
- 5.
See Problem 4.1.1.
- 6.
Only nonnegative fixed points have biological meaning. Given \(R>2\), all three fixed points are nonnegative.
- 7.
Parts (b)–(d) use parameter values adapted from [1].
- 8.
Processes that depend only on the state of a system, and not directly on the time, are called autonomous. Much of the analysis we can do with population models only works for autonomous models.
- 9.
This does not mean that we cannot study the effects of global climate change. We can use analysis to study its effects on fixed points and stability. If we want to study the short-term effects of climate change, however, we need to run simulations.
- 10.
Section 4.1.
- 11.
See [8] for some very unusual patterns in a two-component discrete model.
- 12.
The author’s MATLAB program CobwebPlotter.m provides a convenient way to produce cobweb plots similar to the figures in this section.
- 13.
See Appendix B.
- 14.
This assumes that the dependent variable cannot be negative in the model, as is the case when the dependent variable represents a population. If the model makes sense for negative values of the dependent variable, then use the interval \((-\infty ,\infty )\).
- 15.
- 16.
Section 4.1.
- 17.
Section 4.1.
- 18.
The derivation of this result is in Problem 4.4.16.
- 19.
This section is adapted from [7].
- 20.
The usual lowercase t has been replaced by the uppercase T to allow for systematic use of upper and lower cases in nondimensionalization.
- 21.
Problem 1.4.1.
- 22.
Section 3.2.
- 23.
- 24.
Section 3.6.
- 25.
This is assuming that the phase line analysis is done using the method presented in this section.
- 26.
This is an example of a useful algebra calculation that could not reasonably be done with a computer algebra system. Of course, one could get a CAS to do the calculation by giving it the sequence of steps in the calculation, but not by expecting the CAS to do algebra with human ingenuity.
- 27.
If \(A<1\), then the insect population is not viable on its own, so the questions need not be asked unless \(A>1\).
- 28.
This project is adapted from a model by Maia Martcheva [9].
- 29.
Problem 3.2.10.
- 30.
Note that we’ve incorporated the additional feature into the model by using a dimensionless parameter rather than a dimensional one. We could instead have made the improved recovery rate be \(\gamma +\kappa \), but then \(\kappa \) would need to be scaled.
- 31.
The standard use of phosphates in laundry and dishwashing detergents was linked to lake eutrophication in the late 1960s and spawned one of the early conflicts between the environmental movement and industry. Phosphates are still used in some detergents, but smaller amounts and better treatment of wastewater have significantly reduced their contribution to eutrophication.
- 32.
Carpenter, Ludwig, and Brock estimated \(q=7.8\) and \(r=7.7\) for Lake Mendota, which is adjacent to the campus of the University of Wisconsin at Madison and is probably the most thoroughly studied lake in the world [3].
- 33.
Example 4.4.3.
References
Britton NF. Essential Mathematical Biology. Springer, Berlin and New York (2004)
Butler EP. Pigs is Pigs. American Illustrated Magazine (1905). https://www.gutenberg.org/files/2004/2004-h/2004-h.htm, cited in March 2021.
Carpenter SR, D Ludwig, and WA Brock. Management of eutrophication for lakes subject to potentially reversible change. Ecological Applications, 9: 751–771 (1999)
Iserles A. A First Course in the Numerical Analysis of Differential Equations, 2 ed. Cambridge University Press, Cambridge (2009).
Kruse R, E Schwecke, and J Heinsohn. Heuristic Models. In: Uncertainty and Vagueness in Knowledge Based Systems. Artificial Intelligence. Springer, Berlin, Heidelberg, https://doi.org/10.1007/978-3-642-76702-9_9, 1991.
Ledder G. Scaling for dynamical systems in biology. Bulletin of Mathematical Biology, 79: 2747–2772, 2017.
Ledder G. Qualitative analysis of a resource management model and Its application to the past and future of endangered whale populations. CODEE Journal, 14, http://doi.org/10.5642/codee.202114.01.03 (2021)
Ledder G, R Rebarber, T Pendelton, AN Laubmeier & J Weisbrod. A discrete/continuous time resource competition model and its implications. Journal of Biological Dynamics, 15:sup1, S168-S189, https://doi.org/10.1080/17513758.2020.1862927. (2021)
Martcheva M. An Introduction to Mathematical Epidemiology. Springer, Berlin and New York (2015)
Whiting K. This is how humans have affected whale populations over the years. World Economic Forum, https://www.weforum.org/agenda/2019/10/whales-endangered-species-conservation-whaling/, 2019, cited in December 2020.
Zerbini AN, G Adams, J Best, PJ Clapham, JA Jackson, and AE Punt. Assessing the recovery of an Antarctic predator from historical exploitation. Royal Society Open Science, https://doi.org/10.1098/rsos.190368, 2019.
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Ledder, G. (2023). Dynamics of Single Populations. In: Mathematical Modeling for Epidemiology and Ecology. Springer Undergraduate Texts in Mathematics and Technology. Springer, Cham. https://doi.org/10.1007/978-3-031-09454-5_4
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