Rabbits versus Sheep

In the next few sections we ll consider some simple examples of phase plane analysis. We begin with the classic Lotka-Volterra model of competition between two species, here imagined to be rabbits and sheep. Suppose that both species are competing for the same food supply (grass) and the amount available is limited. Furthermore, ignore all other complications, like predators, seasonal effects, and other sources of food. Then there are two main effects we should consider  

A specific model that incorporates these assumptions is 

To find the fixed points for the system, we solve x = 0 and y = 0 simultaneously. Four fixed points are obtained (0,0), (0,2), (3,0), and (1,1). To classify them, we compute the Jacobian  

This matrix has eigenvalues 2 = -1,-2, as can be seen from inspection, since 

The phase portrait has an interesting biological interpretation. It shows that one species generally drives the other to extinction. Trajectories starting below the stable manifold lead to eventual extinction of the sheep, while those starting above lead to eventual extinction of the rabbits. This dichotomy occurs in other models of competition and has led biologists to formulate the principle of com- 

---

Excised nasal mucosae obtained from various animal species are tools frequently used to study nasal transport and metabolism ([53], Chap. 4). Maintaining the viability of the excised nasal tissues during the experimental period is crucial. Most studies were performed with epithelia excised from rabbits, bovine, sheep, and dogs tissues [54-57], This excised nasal tissue model has been shown to be well suited for studies on nasal permeation and metabolism of drugs. However, species differences in the activity of various enzymes found in human versus these animal nasal mucosae have become an important issue. 

---