Network Science in Food Webs

Introduction

Networks, or graphs, are used as a way to visualize existing information. We create networks to hold a product recommendation system, to visualize interbank loans, or show links between political blogs. There are many instances, like the ones mentioned, where we gain much more insight by taking the sensible approach of using networks in representing information that tends to take the form of relations. There are other instances, however, when we really are just working with relationships, or links, and networks inherently are the underlying structure in real life. One such instance, which will be the focus of the discussion of this blog post, is food webs. We need only visualize the simple, but resourceful network already present, before we can extract our insights and conclusions.

Eat, and be eaten

We should start from the ground up and first look at some simple food webs themselves. Here are two examples:

In Figure 1, we have a rather elementary example of an aquatic food web. It is a graph of nodes connected by directed edges. An edge from Kelp to Small fishes means Kelp is prey for Small fishes. We may now discuss properties about the networks/graphs we see, looking solely through network science lens, and assuming not even the most basic knowledge of biology.

Roots

The example tree/web in Figure 1 has Kelp as its root. Other food webs will have some other organism at the root. For example, in Figure 2, we have grass being the root of a terrestrial food web. What could we gather about the roots of these graphs (remember, we know nothing about biology)? We would run across many more of these examples, and start by trying to find similar characteristics about them, and indeed we would be successful. We would learn that these organisms require sunlight, water, and various other nutrients to thrive. The foundations of these networks, that is, the organisms with no prey, are primary producers; photosynthesis, and chemosynthesis, is their source of energy, not other organisms.

From a network science perspective, we know that even in toy examples where nodes and edges, are simply nodes and edges, trees cannot exist without a root node. We may have algorithms to replace nodes if they are to be deleted when working with toy examples, but in a food web this may not be the case. If grasshoppers, rabbits, and mice did not find grass at their usual feeding ground, they would look elsewhere, but if for some reason they just can’t find any, the food web has no root. This means collapse; the herbivores (which eat plants only) would die out, which means the animals in the next level of the web are also without food, and so on, until this ecosystem vanishes.

Leaves

These nodes have no out-links, this means they are not prey to any other animals. In the food webs presented in Figures 1 and 2, there is only one “leaf” in the graph, but this need not be the case; there may be other animals as leaves, such as in Figure 3. These animals play a special role: “apex predators”. Absolutely no other animal in their ecosystem preys on them, and they also typically do not prey on each other. They may even have the same nodes linking to them (i.e. same prey), but they still manage to co-exist. We will discuss more about how they do so later.

Levels

There are levels which emerge in tree structures, which gives us some motivation to group or categorize the species as they may share behaviours when feeding. There are some simple definitions provided within basic biology to start (see right side labels of Figure 3), but we will look at the approach involving graphs.

In the toy examples of network science, we assign “depth” to nodes in a tree, which is the distance from that node to the root node of the tree. Put another way, starting from the root node, how many edges do we need to follow to get to the target node? The root node will have a depth of 0.

We also know trees/graphs can get more complicated than the one presented in Figure 4, like if a node has more than one in-link, and thus has more than one path to it from the root node. Its depth now depends on which path we take. Similarly, food webs also can get rather complicated, as we see in Figure 3. What we would need to do in such a case is be consistent with how we assign depths. When translating our definitions to better fit food webs, “depth” becomes trophic level, but the definition looks familiar to what we know from our toy examples. Taken directly from Cozzens’ paper:

• Definition – Option 1 The trophic level of species X is:
  1. 0, if X is a primary producer in the food web (a species that does not consume any species in the food web)
  2. k, if the shortest path from a level 0 species to X is of length k.
• Definition – Option 2 The trophic level of species X is:
  1. 0, if X is a primary producer in the food web (a species that does not consume any species in the food web)
  2. k, if the longest path from a level 0 species to X is of length k.

So we will need to choose whether we take the shortest or longest path when determining trophic level.

Weights

If predators share the same prey, how do they co-exist in the same ecosystem? The answer is rather straightforward; they don’t eat the same amount of the same species. What does this mean for our graph? We now have additional information to fit in; the proportions in which predators eat specific prey. This can be represented using weights:

Now, we may think about what this changes in our graph. Perhaps we can hypothesize what would happen if we were to remove snakes, which make up 70% of hawks’ diet. Then they would have no choice but to eat lizards 100% of the time. Foxes will also have to work harder and compete more with hawks to secure lizards to eat. So removing snakes certainly affects its predators (and other predators, which it is not connected with), whom are upstream (i.e. snake out-links to hawk), but of course it also affects animals downstream (i.e. animals linking in to snake), its prey. Rodents and other lizards would have no predators feeding on them, and their populations would grow beyond control, and they would need higher numbers of their prey to feed on. So removing this node has ripple effects on the rest of the web.

Conclusion

Delving deeper into the analysis of food webs also takes us deeper into network science, which is a great tool at our disposal, and a fascinating thing to notice. The study of network science has meant new possibilities in other fields which initially may have been that much more limited in their processes. We only needed to recognize the structure at play of food webs before we were able to use them to our advantage in providing more clarity and rigidity in our analysis of them. It is a great testament as to why network science is important, and why strides in this field also takes many other fields forward.

References