Modeling the pollution of a system of lakes J. Biazar a, * , L. Farrokhi a , M.R. Islam b a Department of Mathematics, Faculty of Science, Giulan University, P.O. Box 1914, P.C. 41938 Rasht, Iran b Faculty of Engineering, Dalhousie University, Suite D-510, 1360, Barrington Street, Halifax, Nova Scotia, Canada B3J 2X4 Abstract Pollution has become a very serious threat to our environment. Monitoring pollution is the first step toward planning to save the environment. The use of differential equations, monitoring pollution has become possible. In this paper, compart- ment modeling is introduced to model the pollution of a system of lakes by a system of differential equations. A system of three lakes, interconnected by channels flowing between them is introduced. Input pollution models are mathematically analyzed and three different types of input models for monitoring the pollution in a lake are introduced. Ó 2005 Elsevier Inc. All rights reserved. 1. Introduction Fig. 1 shows the system of three lakes that are modeled in this study [2]. Each lake is considered to be a large compartment and the interconnecting channel as pipes between the compartments. The direction of flow in the channels or pipes is indicated by the arrows in the figure. A pollutant is introduced into the first lake where p(t) denotes the rate at which the pollutant enters the lake per unit time. The function p(t) may be con- stant or may vary with time. We are interested in knowing the levels of pollution in each lake at any time. We let x i (t) denote the amount of the pollutant in lake i at any time t P 0, where i = 1, 2, 3. We assume the pollutant in each lake to be uniformly distributed throughout the lake by some mixing process. We assume that the volume of water V i in lake i remain constant for each of the lakes. Then the concentration of the pol- lutant in lake i at any time is given by c i ðtÞ¼ x i ðtÞ V i . Each lake initially is assumed to be free of any contaminant, so x i (0) = 0 for each i = 1, 2, 3. To model the dynamic behavior of the system of lakes, we let constant F ji denote the flow rate from lake, i to lake, j. These flow rates, which could be measured in gallons per minute, cubic feet per hour, or any other convenient units, are indicated in Fig. 1. Note that F 12 = 0, since there is no channel allowing any flow from Lake 2 to Lake 1. The flux of pollutant flowing from lake i into Lake j at any time t, denoted by r ji (t), is defined by 0096-3003/$ - see front matter Ó 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2005.11.056 * Corresponding author. E-mail addresses: biazar@guilan.ac.ir, jbiazar@dal.ca (J. Biazar). Applied Mathematics and Computation 178 (2006) 423–430 www.elsevier.com/locate/amc