THE ROSENZWEIG-MACARTHUR PREDATOR-PREY MODEL HAL L. SMITH* SCHOOL OF MATHEMATICAL AND STATISTICAL SCIENCES ARIZONA STATE UNIVERSITY TEMPE, AZ, USA 85287 Abstract. This is intended as lecture notes for 2nd ODE course, an application of the Poincar´ e-Bendixson Theorem. The model is derived and the behavior of its solutions is discussed. 1. The Model We derive and study the predator-prey model which Turchin [7] at- tributes to Rosenzweig and MacArthur [8]. Turchin’s book is an ex- cellent reference for predator-prey models. See also Hastings [2] and Murray [6]. Let x denote prey density (#/ unit of area) and y denote predator density (#/unit of area). A typical predator-prey model has the form x ′ = birth rate - death rate not due to y - kill rate due to y y ′ = -death rate + reproduction rate Models differ in the choices made for these. Logistic growth and death is a common choice made to model prey birth and death in the ab- sence of predators. A linear death rate for predators is common. If predator density is not so large that they interfere with each other while searching for prey, then one often assumes that the death rate due to predators is linear in predator density. Also, it is common to assume that predator reproduction rate is proportional to the predator kill rate. Thus, we have x ′ = rx  1 - x K  - yh(x) y ′ = y (-c + dh(x)) where r, K > 0 and c, d > 0. Note that h(x) has units of 1/time: h(x) = #prey caught per predator per unit time. It is the per predator kill rate. 1