Modelling the Relationship between Resource Availability and Depletion using Ordinary Differential Equations

Modelling the Relationship between Resource Availability and Depletion using Ordinary Differential Equations

Most of the world’s resources are nonrenewable. While this is a widely accepted fact, the 21stcentury presents unique challenges in terms of balancing demand for resources such as water, fish, and land with sustainability requirements. Seo (18) states that the effects of climate change have only made the situation worse, especially as rainfall patterns are becoming increasingly unpredictable, water sources are becoming depleted, forest cover is declining, the acreage of arable land is shrinking, and fish populations are dwindling. Marin and Ochsner (37) argue that while all this is happening, the global population continues to grow, raising serious questions over the planet’s ability to support itself.

Is there a way that the correlation between resource utilization and sustainability can be analyzed, predicted, and modeled to enable better resource planning and management? At what point can available resources become incapable of keeping up with consumption? These questions – and more – can be answered by mathematics as it is one of the few disciplines whose applications are virtually limitless. According to Schiesser (33), ordinary differential equations (ODE), which have been found to be particularly relevant in solving real-life problems, are applicable in this context.

Marin and Ochsner (13) content that mathematically, the relationship between resource availability and resource consumption can be modeled using the following differential equation: dP/dt=kP(1-P/N)-λ. In this formula, the population (resource abundance/availability) is represented by P(t), the growth rate of the resource is denoted by k, N symbolizes rate of resource consumption, and the carrying capacity is signified by λ. An examination of the answers to this equation for different resource consumption levels confirms that a critical resource consumption level, also referred to as a bifurcation value, is a reality(Marin and Ochsner 15).

The equation reveals that if the resource consumption level surpasses this bifurcation value by the smallest degree, then the specific resource – fish, trees or water among others – will dramatically reduce to the extent that it becomes wholly or almost depleted (Schiesser 49). The logic behind the equation is that assuming a region, country, society or community is approaching the bifurcated value, the slightest increase in the consumption rate can have devastating effects on the availability of the resource (Schiesser 68). As a consequence, great caution should be exercised when expanding resource utilization even by minute quantities; failure to do this means it will cease to exist at some point.

A community that grazes its cattle on a common piece of land, for instance, may not take any action to moderate the intensity of grazing that the cattle are allowed to engage in. After some time, pastures on the land will become extinct and unable to sustain any further grazing; in other words, the population of the pasture will crash (Schiesser 61). Overutilization of resources leading to their collapse and the extinction of whole societies or communities also referred to as overharvesting, is not a new phenomenon. Indeed, there are historical accounts that prove it should be treated as a matter of urgency. According to Seo (29), a good example is the deterioration of the community on Easter Island, home to the Rapa Nui people. Credible documents show that a critical ingredient in the society’s disintegration was the total depletion of forest cover on the island, made possible by human deforestation.

Today, many would wonder how naïve the Rapa Nui were to decimate all trees in their habitat. How did no one realize that the forest cover was rapidly disappearing? Why did no one intervene to arrest the situation? These questions are easy to pose now, but back then the Rapa Nui were oblivious to the level of destruction they were causing to themselves and their environment. Seo (57) adds that ironically, while it is easy for current generations to feel arrogantly smarter than the Rapa Nui, it is likely that future generations will ask the same questions regarding actions that are presently causing the extinction of species and overconsumption of food sources.


Works Cited

Marin, Marin, and Ochsner, Andreas. Essentials of Partial Differential Equations: With

Applications. Berlin: Springer International Publishing, 2018.

Schiesser, William E. Differential Equation Analysis in Biomedical Science and

EngineeringPartial Differential Equation Applications with R. New York, NY: Wiley, 2014.

Seo, Niggol S. Natural and Man-Made CatastrophesTheories, Economics, and Policy Designs.

New York, NY: John Wiley & Sons.

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