Equation of the Month

Equation of the Month

A blog run by the

Theoretical Population Ecology and Evolution Group,

Biology Dept.,

Lund University

The purpose of this blog is to emphasize the role of theory for our understanding of natural, biological systems. We do so by highlighting specific pieces of theory, usually expressed as mathematical 'equations', and describing their origin, interpretation and relevance.

Thursday, February 10, 2011

The Fear Equation

where μ is (perceived) predation risk, F is current fitness, and ∂F / ∂e is the marginal fitness gain from acquiring more energy from foraging.

What does it mean? Technically speaking, the equation represents the marginal rate of substitution of safety for food. It says that when the environment is risky (high rates of predation), when current fitness is high (e.g., if the animal is well fed), and when the marginal gain of more food is low, then one should be very risk averse, i.e., feel “fear”.

Where does it come from? It originates from Joel Brown’s seminal paper (Brown 1988) formulating the relationship between patch use, foraging rate and predation risk.

Importance: This idea has subsequently been much explored when studying foraging ecology, habitat selection, and the mechanisms of coexistence between competitors and predators and their prey. It elegantly shows how different fitness “currencies” (here, food and safety) can be translated into each other. This trick is often necessary when putting together reliable and realistic fitness functions for many problems in evolutionary ecology. It also determines the “landscape of fear” prey populations experience and it can be shown that this effect on the population can be greater than the actual killing of prey individuals. It also nicely explains the “Stalingrad effect”, i.e., the fearless behavior of the inhabitants of the city during the WWII battle under severe risk. They had with very low current “fitness” and extremely high marginal “fitness“ gain from some food. Think about similar situations yourselves!

Per Lundberg


Brown, J.S. 1988. Patch use as an indicator of habitat preference, predation risk, and competition. Behav. Ecol. Sociobiol. 22: 37-47.

Brown, J. S. 1992. Patch use under predation risk: I. Models and predictions. Ann. Zool. Fennici 29:301-309.

Brown, J. S. & Kotler, B. P. 2004. Hazardous duty pay and the foraging cost of predation. Ecol. Lett. 7: 999-1014.


  1. So it seems you have more to fear when you are fitter, which makes sense.

    So given the equation from Star Wars (Yoda) Fear=Anger=Hatred - then this explains why rich (fit) people get so angry (fearful).

    Or - the corollary - conservatives are susceptible to the politics of fear.

    But, why are old people so frightened, they should be bolder, they have so little to lose.

    Cheers Hugh Possingham

    PS - Joel Brown is a genius

  2. Dear Hugh,
    Thanks for thoughtful comments on the "fear equation".

    My take on the old folks paradox is that the have high mu. They perceive the environment as dangerous. And it is, in a way. Old people aren't very strong and fast so threats that others would find trivial, they might be (rightfully) afraid of.

    We have done a preliminary analysis of some attitude data in Sweden and there might be a signal when it comes to rich people. Rich people tend to perceive the same risk (nuclear fallouts or robbery or car accidents) as mor "fearful" than not so wealthy people.

    Per L.

  3. Thanks for inviting me to this exclusive club!

    Criminological data sets can perhaps be used since they measure "fear" in different ways and have lots of background variables such as income, sex, age, und so weiter.
    Only in Swedish, unfortunately...
    Mikael Sandberg

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