Emperor Penguin Energy-Risk Model - Part 2

Emperor Penguin Diving onto Ice Shelf from Sea, Stancomb Wells Ice Edge, Weddell Sea, Antarctica (Image on Alamy.com)

In my last blog post “An Emperor Penguin Energy-Risk Model” on April 14, 2011, l discussed the predator-prey relationship between the leopard seal and emperor penguin in Antarctica. The leopard seal waits at the edge of the ice shelf and opportunistically picks off emperor penguins entering or leaving the sea. For the emperor penguin, feeding at sea is a decision between the need to feed to live and the risk of dying in the mouth of a leopard seal..

In the blog post I state: “From studying the emperor penguin and the leopard seal we know the emperor penguins will continue to feed, but so will the leopard seal. Some emperor penguins, despite their various risk protection strategies, will get eaten. It is important to note that in a probabalistic sense, we know that some penguins will be eaten by the leopard seal, but we don’t know which specific penguins will “bite the dust”.” This is true casually looking at a row of emperor penguins lined up to go into the sea in search of food.

However, upon closer analysis and study, over a period of time, it might be possible to determine which emperor penguins have a bit of catch in their step, have been injured in a narrow escape from a leopard seal, or have slowed down. These emperor penguins might come a belly-flop short of landing on the ice, and end up as prey in the mouth of a leopard seal. However, it is also possible, that a healthy, fit, member of the emperor penguin colony might suffer a particularly ill-fated episode of bad luck. This penguin might be in the wrong place at the wrong time when the leopard seal is rising out of the water with its mouth wide open ready for business. In fact, you could have the emperor penguin equivalent of the 4.0-40 yard dash champion, and end up as leopard seal “dinner”, with some bad luck and timing.

Looking at emperor penguin energy-seeking behavior and risk, it becomes apparent that probabilities have a great deal to do with the outcome but are not deterministic. You may attach a relatively higher probability of being eaten to the more fragile members of the emperor penguin population and a relatively lower probability of being eaten to those fitter members. The larger the colony size, and the more emperor penguins entering the sea at the same time, the lower the risk, the probability of being eaten, for any particular emperor penguin as there are more penguins entering the sea. (“there’s safety in numbers”).

You can run scenarios with differing proportions of fragile and fit emperor penguins, with higher and lower probabilities of being eaten (mortality rates), varying degrees of illness (morbidity rates) or accident, including leopard seal attack. In such scenarios, the leopard seal would most likely pick off different emperor penguins each time the scenario is run, however there would be objective tendencies to pick off more members of the more fragile group versus those of the fitter group.

In performing mathematical modeling of the fate of the emperor penguins by running scenarios with objective data and assumptions, we may set up a stochastic process which helps us to understand the behavior of the system as it evolves under a variety of scenarios.

Mathematical models involve expressing real world problems in mathematical language. This entails defining variables and establishing a formulaic process which will express the model as evolves. Variables are elements in the model which may change during the model. Because they may change, the model needs to calculate how they change over the course of the model and how they interact with other model variables, and are affected by the constants assumed by the model. Constants may arise from established data or may be assumptions plugged in to the model.

Stochastic processes incorporate non-deterministic, random elements into a mathematical model. The result may vary with time and with each model run. In comparison, a deterministic model will always produce the same result given the same assumptions and initial state. Thus, a stochastic process is run using random processes, employing a variety of assumptions and probability distributions informing objective tendencies for various model events to occur..

The random process in stochastic modeling will randomly choose which penguins are attacked, survive, suffer morbidity or injury from accident, and die over a period of time. Each run will be unique, as specific, members of the colony are differently impacted by the random process each time. By running many such models, one can get a picture of the survival data for the colony as a whole under a wide range of assumptions. Depending on the characteristics of the data, model and variables, results may be similar on an overall group basis, while differing by individual members impacted over time.

Under a normal range of assumptions and outcomes, this model may well predict overall group behavior over a period time. However modeling becomes much challenging when very low probability events enter into the model or rear their head in actual life.

For example, a eruption of an Antarctic volcano may be infrequent, however it could certainly impact emperor penguins. If the model assumed a volcanic eruption with a low probability, a robust number of stochastic model runs may randomly select such an event resulting in a BBQ penguin supper for the leopard seals.

The objective of the stochastic processes is to help us inform our decision making process, to help us understand the impact of variables under a wide range of assumptions, conditions, and scenarios. Thus a stochastic process should inform us about the expectations of the model under a wide variety of conditions, including the impacts of low probability / high risk events.