Biological systems are complex to model mathematically, but optimization theory grounded in Darwin's evolution offers a solution. This approach predicts organism behaviours under various conditions, aiding in understanding responses to changes like climate shifts. Interdisciplinary collaboration between mathematicians and ecologists is key to advancing this research. Read more in the blog post by Associate Professor Uffe Høgsbro Thygesen from DTU Compute.
Text: Associate Professor Uffe Høgsbro Thygesen from DTU Compute. Photo: Sebastian Pena Lambarri on Unsplash
Biological systems are notoriously difficult to describe mathematically, because we do not have a well established set of equations of motion, like those we have in physics. In a situation where life sciences take on a larger and larger role in society as well as on university campuses, this presents a challenge to applied mathematicians, but also offers an opportunity.
The problem is that a biological organism has a vast number of degrees of freedom, and their dynamics are constrained by the laws of physics - such as conservation of mass and energy - but this is in no way enough to define the dynamics.
We often have semi-empirical models for things like how the growth of organisms respond to changes in environment, but they typically only have limited applicability.
An example where this becomes critical is the prediction of climate change. This field involves numerous feedback between the physical environment and the behaviours of organisms. But what will organisms do when they experience global change; which processes will accelerate, which organisms will decline in abundance, and who will move?
One theory, more than 150 years old, provides part of the solution: Darwin's theory of Evolution by natural selection. It is an extremely strong guiding principle. As Dobzhansky wrote, "nothing in biology makes sense except in the light of evolution"; every feature of every organism is the result of a million of years of fitness-maximizing natural selection.
From a mathematical point of view, this suggests posing an optimization problem which predicts the fitness of an organisms as a function of all those parameters, which we don't know. We then solve this optimization problem and assume that nature, over evolutionary time scales, has selected the same solution. If the model gives accurate predictions, then we have some confidence that we have understood what the organism has been selected to optimize, and which constraints it is operating under. This allows us to predict how it will respond to changes.
Personally, I was introduced to this paradigm in ocean science, where it has given startling results. One phenomenon which impressed me was that when we catch more fish, the fish respond by changing their life history so that they reproduce earlier in life, at a smaller size. This is an example of a feedback in the ecosystem, which is crucial for the dynamics of the system, and which is obvious only from an evolutionary point of view.
While evolution is not a silver bullet, this presents research opportunities for applied mathematicians to collaborate with theoretical ecologists.
A key question is what exactly is fitness. We can often understand it as the expected number of descendants of an individual, but this criterion may be difficult to make operational. I have studied the phenomenon of vertical migrations of oceanic organisms, where we argue for a trade-off between energetic gains and death risks. In other situations, it is critical that we understand fitness as a property of the gene, not of the individual.
A final consideration is that fitness is a property of an individual (or a gene), but depends on its environment, including all other individuals (or genes), which also are under selection. This means that it, sometimes, is important to cast the optimization problems in terms of game theory.
We applied this notion to the study of animal migrations and used dynamic optimization theory to show that the animals, collectively, satisfy equations which are similar to those that govern fluid flow. This was a satisfying result, because it uses individual-based arguments to predicts system-level outcomes such as spatial distributions and population dynamics.
Biological systems are complex to model mathematically, but optimization theory grounded in Darwin's evolution offers a solution. This approach predicts organism behaviour under various conditions, aiding in understanding responses to changes like climate shifts. Interdisciplinary collaboration between mathematicians and ecologists is key to advancing this research.
Associate Professor Uffe Høgsbro Thygesen from DTU Compute. Photo: Hanne Kokkegård, DTU Compute
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