Theories of optimality involve a mathematical model of cost and benefit analysis that can give quantitative predictions about an animal's behavior. These hypotheses can be tested experimentally or through comparative phylogeny. By no means are these models fully accurate; each comes with certain assumptions and variations that are not included in these simple models. But we can often use the models to predict behavior to a reasonably certain degree.
To some degree, animals display the ability to modify their behavior so that they receive an optimal balance of benefits and costs. Costs can include danger, loss of valuable time, and wasted energy. Benefits are usually counted in terms of net energy intake (consumed calories) per unit time or number of offspring produced. In this section, we will focus on optimal foraging methods to achieve the highest net energy intake. We will discuss reproductive fitness and the production of offspring in the next section.
Contingency Theory, also called the prey choice model, predicts what an animal will do when it encounters a particular food. Should the animal eat what he has, or search for a more profitable food item? We do not often imagine animals refusing to eat the food in front of them to search for other items, but this does occur. Shorecrabs, for instance, eat muscles that become increasingly difficult to crack open as their size increases. Crabs will pass up large muscles, which would take too much time and energy to crack, to search for smaller muscles. This way, the crab can spend less time and energy handling their food, and, even though they pass up the massive meals, will increase their net food intake.
Models similar to that just described for the Shorecrab can include several food choices, though the math involved can quickly complicated; we will focus on the simpler version with two food types.
PARARAPH For the purpose of the model, we will define the following terms:
Food choice 1 is scarce, but is highly profitable, meaning it will yield a high amount of energy with a low handling time. E/h for food 1 is therefore quite high. However, because we are trying to maximize both E/h and E/t, we must also take into account the time it takes to find the very scarce food choice 1.
Food choice 2 is abundant, but less profitable than food 1. E/h for food source 2 is not very high, but it takes much less effort and time for the animal to find food choice 2.
The model assumes the animal is holding food 2, meaning that there is no search time involved for food choice 2 since the animal has already found it. The animal stands over the food and must debate whether to eat it: is the immediate consumption of food choice 2 a better action than moving on and looking for some of that fine food choice 1? We can put this debate into mathematical terms:
If E2/h2 > E1/(s1 + h1) then the animal should eat food 2.If the profitability of food choice 2 is greater than the energy of food choice 1 divided by the sum of the search and handling time of food source 1, then eating food 2 is the better move. If the energy per time gained by going in search of food source 1 is higher, then the animal should pass by food choice 2 and keep searching for food type 1.
Think about the problem posed if the animal was standing above food choice 1 rather than food choice 2. Because food type 1 is more profitable, the animal should always eat it if it comes upon it. Therefore, for the model's purposes, We only consider food type 2 because type 1 is hard to come by.
From the model for contingency theory, we can see that inclusion of a food type in an animal's diet is dependent only on the abundance of better food choices, and is independent of that food type's own abundance. The model predicts that when all food types are abundant, diets are restricted to fewer types, because the animal can afford to be choosier. With this model, we can often predict an animal's optimal diet. However, the animal itself will not always be able to predict his own ideal diet because the model assumes the animal has perfect knowledge of available resources. In order to know the benefits of two food types, the animal must consume both and observe the relative abundance of both types. And so, what we see in nature does not follow the model exactly, but it does come close.
Marginal value theory, also called patch choice theory, is a form of the economic law of diminishing returns. An animal feeding at a food patch must decide when to leave the patch in search of another. The more of the patch the animal consumes, the lower the rate of return will be for the remainder of the patch because the food supply is running out. Using calculus, we can determine the optimal time for the animal to leave the patch and search for a new one. When the profitability of the patch lowers enough to equal the profitability of an average patch, including the time it will take to search or travel to the new patch, the animal should leave. Mathematically, the optimal time to leave is: dE(h)/dh = E(h)/(s+h). You should be aware this formula exists, but need not know how to use it. There is a simpler, graphical method for determining the optimal time to spend at any one patch.
As we can see in the , the rate of calorie consumption decreases as the animal spends more time at one patch (the slope of the graph decreases). The total calories continues to increase, but the animal would benefit more by finding a fresh patch from which the rate of consumption would be higher.
The easier to use graphical method is illustrated in . If we know how long it will take on average to find a new patch, we draw a line from that search time so that it is tangent to the curve. The point at which our line touches the curve is the optimal time to spend at the patch. After this time, the animal should leave.
The marginal value theory predicts that animals will remain in one patch longer when patches are scarce or far apart. Like the contingency model, the patch choice model is not perfect; it merely serves to approximate the amount of time an animal will spend in any one patch.