Mathematical modeling and analysis of the depletion of dissolved oxygen in water bodies

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1 Nonlinear Analysis: Real World Applications Mathematical modeling and analysis of the depletion of dissolved oxygen in water bodies A.K. Misra a, Peeyush Chandra a,, J.B. Shukla b a Department of Mathematics, Indian Institute of echnology, Kanpur 20806, India b Centre for Modeling Environment and Development, Kanpur , India Received 4 January 2005; accepted 5 September 2005 Abstract In this paper, a nonlinear mathematical model is proposed to study the depletion of dissolved oxygen in a water body caused by industrial and household discharges of organic matters pollutants. he problem is formulated as a food chain model by considering various interaction processes biodegradation and biochemical involving organic pollutants, bacteria, protozoa, an aquatic population and dissolved oxygen. Using stability theory, it is shown that as the rate of introduction of organic pollutants in a water body increases, the concentration of dissolved oxygen decreases due to various interaction processes. It is found that if the organic pollutants are continuously discharged into water body, the concentration of dissolved oxygen may become negligibly small, thus threatening the survival of aquatic populations. However, by using some effort to control the cumulative discharge of these pollutants into the water body, the concentration of dissolved oxygen can be maintained at a desired level Elsevier Ltd. All rights reserved. Keywords: Mathematical model; Water pollution; Dissolved oxygen; Monod interaction. Introduction It is well known that pollutants and toxicants discharged in water bodies are harmful to biological species such as a fish population 0,5,9,22,23. When household and industrial wastes are discharged into water, the organic matter present in them are uptaken by micro-organisms which convert them into inorganic matter using dissolved oxygen in the process. he density of these micro-organisms bacteria increases as the cumulative concentration of organic pollutants increases. As these are part of a food chain involving protozoa and other biological populations in a water body, the level of dissolved oxygen decreases further due to various interactive biochemical and biodegradation processes. It may be noted here that the input of the dissolved oxygen in a water body is mainly due to atmospheric diffusion through the water surface and to a certain extent due to its production by photosynthesis 3,0,8 20. herefore, while modeling the depletion of dissolved oxygen in an aquatic ecosystem, all these processes should be taken into account 9. he study of effect of the organic pollutants on dissolved oxygen in a water body started with the well-known model of Streeter and Phelps 2. Several workers including Dobbins 3, O Connor 4, Beck and Young have generalized this model. It may be pointed out here that these models are only linear. However, there exist various nonlinear Corresponding author. address: peeyush@iitk.ac.in P. Chandra /$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:0.06/j.nonrwa

2 A.K. Misra et al. / Nonlinear Analysis: Real World Applications processes in water bodies involving interactions such as organic pollutants with bacteria 6,,5, phytoplankton with zooplankton in presence of nutrients 2,8, organic chemical distribution in aquatic food chains 22, dissolved oxygen with aquatic populations 5, etc. For a case related to organic pollutants, Rinaldi et al. 5 proposed ecological type nonlinear models involving fast and slow biodegradable pollutants, which implicitly consider the depletion of dissolved oxygen by bacteria and protozoa in a water body. In this study, where simulation analysis has been carried out, several questions regarding the behavior of the nonzero equilibrium including nonlinear stability, etc. remained unanswered. Further some other aspects, such as natural depletion rates of pollutants, crowding of species, growth of oxygen from algae and macrophytes, survival of aquatic population dependent on dissolved oxygen, etc. have not been taken into account in various nonlinear modeling studies. It may be also pointed out here that, the level of dissolved oxygen can be increased by pumping air into the water body, being used for production of fish, and therefore, this aspect needs also to be considered in the modeling process. In this paper, therefore, a nonlinear ecological type of mathematical model for the depletion of dissolved oxygen in a water body is proposed and analyzed by considering some of the aspects mentioned above involving degradable organic pollutants, bacteria, protozoa and an aquatic population. his model takes into account monod interactions between organic pollutants and bacteria, bacteria and protozoa and a bilinear interaction between protozoa and the aquatic population. It is assumed that a constant rate of input of oxygen is applied into the water body, to increase the level of dissolved oxygen 3,8,9. A model to control the discharge of organic pollutants and its effect on dissolved oxygen is also proposed and analyzed. 2. Mathematical model We consider a water body where organic pollutants are discharged in the form of wastes. It is assumed that these pollutants are part of a food chain consisting of bacteria, protozoa and an aquatic population, for example a fish population using dissolved oxygen in this water body for various biochemical and biodegradation processes. Let, be the cumulative concentration of organic matter degradable pollutants, B be the density of the bacteria, P be the density of the protozoa, N be the density of an aquatic population, which depends wholly on protozoa and C be the concentration of the dissolved oxygen DO. It is assumed that the cumulative discharge of organic pollutants into the water body is at a constant rate Q and that the rate of decrease of concentration due to biochemical and other factors is proportional to. It is further assumed that the cumulative rate of depletion of due to bacteria is given by a monod type of interaction involving the density of bacteria B as well as the concentration i.e. B/K 2 + K. As bacteria wholly depend upon organic pollutants, the growth rate of density of its population is proportional to this term. It is considered that the natural depletion rate of density of bacteria is proportional to B and the decrease of its growth rate due to crowding is proportional to B 2. Further, the depletion rate of density of bacteria by its predator protozoa is assumed to be given by the monod interaction between bacteria and protozoa i.e. BP/K 2 + K 22 B. hus, the growth rate of protozoa density is also proportional to this interaction term. he natural depletion rate of protozoa density is assumed proportional to P while its decrease due to crowding is proportional top 2. he depletion rate of protozoa density by its predator such as a fish population is assumed proportional to PN and therefore the corresponding growth rate of density of this predator population is proportional to this product. he natural depletion rate of density of predator population is considered proportional to N and its decrease due to crowding is proportional to N 2. It is also assumed that the rate of growth of concentration of dissolved oxygen by various sources including diffusion, pumping of air into the water body, etc. is q a constant and its natural depletion rate is proportional to C.Itis assumed, further, that the rate of depletion of dissolved oxygen is proportional to various terms K B/K 2 + K, K 2 BP/K 2 + K 22 B, K 3 PN, α B,α 2 P,α 3 N representing growth and depletion rates 5. Keeping in mind the above considerations, the system is governed by the following differential equations: d db = Q α 0 K B K 2 + K, K B = λ K 2 + K α B λ 0 B 2 K 2 BP K 2 + K 22 B,

3 982 A.K. Misra et al. / Nonlinear Analysis: Real World Applications dp dn dc = λ 2 K 2 BP K 2 + K 22 B α 2P λ 20 P 2 K 3 PN, 2. = λ 3 K 3 PN α 3 N λ 30 N 2, K B = q α 4 C λ 34 K 3 PN λ α B λ 22 α 2 P λ 33 α 3 N λ 2 K 2 + K λ K 2 BP 23 K 2 + K 22 B, where 0 = 0 0, B0 = B 0 0, P0 = P 0 = P 0 0, N0 = N 0 0, C0 = C 0 0. In model 2. the constants λ 0, λ 20, λ 30 are crowding coefficients flaking off coefficients 5 of bacteria, protozoa and the aquatic population, respectively. he parameters K,K 2,K 3,K,K 2,K 2,K 22 involved in the monod type interactions are all constants. he coefficients λ, λ 2, λ 3, λ, λ 22, λ 33, λ 2, λ 23, λ 34 are positive constants and their magnitudes are less than equal to unity. It may be pointed out that for feasibility of model 2., the growth rates of bacteria and protozoa should be positive. Hence, from the second and third equation of the model, it follows that and λ K K α > λ 2 K 2 K 22 α 2 > Following, Freedman and So 5, the region of attraction Ω for all solutions initiating in the positive octant is given by } Ω = {0 Qα0, 0 B R B, 0 P R P, 0 N R N, 0 C qα4, 2.4 where R B = R P = λ K Q λ 0 K Q + K 2 α 0, 2.5 λ λ 2 K K 2 Q λ 20 K 22 λ K Q + λ 0 K 2 K Q + K 2 α 0, 2.6 λ λ 2 λ 3 K K 2 K 3 Q R N = λ 20 λ 30 K 22 λ K Q + λ 0 K 2 K Q + K 2 α It is assumed here that all the initial values of the variables belong to set Ω. In the following, we analyze this model 2. by using the qualitative theory of differential 4,7,2,6,7. 3. Equilibrium analysis System 2. has following four nonnegative equilibria in Ω. hey are listed below: i E Q/α 0, 0, 0, 0,q/α 4 which always exists. ii E 2 2,B 2, 0, 0,C 2 exists, provided the following conditions are satisfied: λ K K α Q K 2 α 0 α > 0, 3. q λ 2 K 2 B 2 K 2 + K 2 λ α B2 >

4 A.K. Misra et al. / Nonlinear Analysis: Real World Applications iii E 3 3,B 3,P 3, 0,C 3 exists, provided the following conditions are valid: λ 20 λ K K α Q K 2 α 0 α + K 2α 2 K Q + K 2 α 0 >0, K λ 0 + K 2 λ 2 K 2 K 2 λ2 K 2 R B λ 20 K 2 + K 22 R B 2 K 22 α 2 > 0, K 2 + K 22 R B K 2 + K 22 R B 3.4 λ 2 K 2 B 3 K 2 + K 22 B 3 α 2 > 0, 3.5 q λ 2 K 3 B 3 K 2 + K 3 λ 23 K 2 3 B 3 K 2 + K 22 B 3 λ α B3 λ 22α 2 P3 > iv E 4,B,P,N,C exists whenever the following conditions are satisfied: λ K K α Q K 2 α 0 α + K 2m K 2 l K Q + K 2 α 0 >0, 3.7 K 2 λ30 λ 2 K 2 K 2 λ 0 + K 2 + K 22 R B 2 l K 2 + K 22 R B K λ30 λ 2 K 2 R B 22 K 2 + K 22 R B m > 0, 3.8 λ 30 λ 2 K 2 B K 2 + K 22 B m>0, 3.9 λ 3 K 3 P α 3 > 0, 3.0 K B q λ 2 K 2 + K λ K 2 B P 23 K 2 + K 22 B λ 34K 3 P N λ α B λ 22 α 2 P λ 33 α 3 N > 0, 3. where l = λ 20 λ 30 + λ 3 K 2 3, m = λ 30α 2 K 3 α 3. Proof. he equilibria E i i =, 2, 3, 4 of system 2. are obtained by solving the following set of simultaneous algebraic equations: K B Q α 0 = 0, 3.2 K 2 + K λ K B K 2 + K α B λ 0 B 2 K 2 BP = 0, 3.3 K 2 + K 22 B λ 2 K 2 BP K 2 + K 22 B α 2P λ 20 P 2 = 0, 3.4 λ 3 K 3 PN α 3 N λ 30 N 2 = 0, 3.5 K B q α 4 C λ 2 K 2 + K λ K 2 BP 23 K 2 + K 22 B λ 34K 3 PN λ α B λ 22 α 2 P λ 33 α 3 N = he equilibrium E Q/α 0, 0, 0, 0,q/α 4 exists obviously. We show the existence of other equilibria as follows. Existence of E 2 : for the equilibrium E 2 2,B 2, 0, 0,C 2, the values of 2,B 2 and C 2 are obtained by solving the following algebraic equations: B = Q α 0 K + K 2, 3.7 K B = λ K λ 0 K 2 + K α. 3.8

5 984 A.K. Misra et al. / Nonlinear Analysis: Real World Applications B Fig.. Existence of 2,B 2. From the isocline given by 3.7, it is easy to note the following: i B = 0at = Q/α 0 ii B as 0, iii db d = K 2 Q α 0 2 K α 0 K K + K 2, which is negative for 0 Q/α 0. hus B decreases as increases. From the isocline given by 3.8, it is noted that B<0at = 0 and B = 0at = K 2 α /λ K K α, which is positive. Further db/d >0 and hence B increases for all >0. hus, the two isoclines 3.7 and 3.8 intersect at 2, B 2 see Fig. provided λ K K α Q K 2 α 0 α > 0. Using these values of 2 and B 2,wegetC 2 from 3.6 as follows: C 2 = α 4 q λ 2 K 2 B 2 K 2 + K 2 λ α B 2, 3.9 which is positive provided the right-hand side of 3.9 is positive. his gives condition 3.2. hus, E 2 2,B 2, 0, 0,C 2 exists, provided conditions 3. and 3.2 are satisfied. Existence of E 3 : for the equilibrium E 3 3,B 3,P 3, 0,C 3, the values of 3 and B 3 are given by the following algebraic equations: B = Q α 0 K + K K and λ K K 2 + K α = λ 0 B + K 2 K 2 + K 22 B λ 20 λ2 K 2 B K 2 + K 22 B α Isocline 3.20 is the same as 3.7, hence its behavior is also the same. From the isocline given by 3.2, it is noted that B increases with in 0 B R B R B is given by 2.5, provided inequality 3.4 is satisfied. Also B = 0at = K 2 α K / 2α 2 K 2 α 2 λ K K α + K, λ 20 K 2 λ 20 K 2

6 A.K. Misra et al. / Nonlinear Analysis: Real World Applications B Fig. 2. Existence of 3,B 3. which is positive or negative depending on the sign of ᾱ = α K 2 α 2 /λ 20 K 2. It is noted that the value of B at = 0 is given by the following cubic equation: λ 0 λ 20 K 2 22 B3 + 2λ 0 λ 20 K 2 K 22 + λ 20 K 2 22 α B 2 + λ 0 λ 20 K K 2λ 2 K 2 K 22 α 2 + λ 20 α K 2 K 22 B + λ 20 K 2 2ᾱ = It is clear that one root of Eq is positive or negative depending on whether ᾱ is negative or positive, respectively. hus, from the above analysis we conclude that the two isoclines 3.20 and 3.2 intersect at a unique point 3,B 3 in the interior of first quadrant see Fig. 2, which is drawn for ᾱ > 0 provided conditions 3.3 and 3.4 are satisfied. Using these values of 3,B 3 in Eqs. 3.4 and 3.6, we get P3 = λ2 K 2 B3 λ 20 K 2 + K 22 B3 C3 = K 3 q λ B 3 2 α 4 K 2 + K 3 α 2, 3.23 λ 23 K 2 B 3 P 3 K 2 + K 22 B 3 λ α B 3 λ 22α 2 P which are positive under conditions 3.5 and 3.6, respectively. hus E 3 exists provided conditions are satisfied. Keeping in mind model 2., it is noted on physical consideration that the growth rate of bacteria in the absence of protozoa is more than when protozoa is present as protozoa predate bacteria. hus B2 >B 3. his result can also be proved from model 2. by using a comparison theorem 7. Existence of E 4 : for the equilibrium E 4,B,P,N,C, and B are obtained by solving the following algebraic equations: B = Q α 0 K K + K and λ K K 2 + K α K 2 λ30 λ 2 K 2 B λ 0 B K 2 + K 22 B l K 2 + K 22 B m = 0, 3.26 where l = λ 20 λ 30 + λ 3 K3 2 and m = λ 30α 2 K 3 α 3.

7 986 A.K. Misra et al. / Nonlinear Analysis: Real World Applications B Fig. 3. Existence of,b. Isocline 3.25 is the same as 3.7; hence, its behavior is also the same. From isocline 3.26, it is noted that under condition 3.8 B increases with in 0 B R B. It is also noted that B = 0at = K 2 α K / { 2m λ K K α + K } K 2 m K 2 l K 2 l which is positive or negative depending upon the sign of α = α K 2 /K 2 m/l. Further, for this isocline, the value of B at = 0 is given by the following cubic equation: λ 0 K22 2 B3 + 2λ 0 K 2 K 22 + K22 2 α B 2 + λ 0 K2 2 + K 2 λ 30 λ 2 K 2 mk K 2 K 22 α B l + K2 2 α = One root of Eq is positive or negative depending on the negative or positive value of α, respectively. hus, from the above discussion, we conclude that the two isoclines 3.25 and 3.26 intersect at a unique point,b in the interior of first quadrant see Fig. 3, drawn for α > 0 provided 3.7 and 3.8 are satisfied. Using these values of and B we get, P = λ30 λ 2 K 2 B l K 2 + K 22 B m, 3.28 N = λ 3 K 3 P α 3, 3.29 λ 30 C = K B q λ 2 α 4 K 2 + K λ K 2 B P 23 K 2 + K 22 B λ 34K 3 P N λ α B λ 22 α 2 P λ 33 α 3 N 3.30 which are positive under conditions hus E 4 exists, provided conditions are satisfied. On physical consideration from model 2. it is noted here that growth rate of protozoa in the absence of lower order aquatic population is greater than that in its presence as protozoa is predated by this aquatic population. Hence P 3 >P. his result can also be proved from model 2. by using a comparison theorem 7.

8 A.K. Misra et al. / Nonlinear Analysis: Real World Applications Remark.. Using Eqs and 3.26 we can check that under condition 3.8, d /dq>0 and db /d > 0. his gives db /dq = db /d d /dq > 0. Also, from Eq. 3.28, we note that dp dq = l λ 30 λ 2 K 2 K 2 db K 2 + K 22 B 2 dq > 0. Similarly from Eq. 3.29, we get dn /dq = λ 3 K 3 /λ 30 dp /dq > 0. Hence,B,P and N increase as Q increases. Also from Eq it can be seen that dc /dq<0, dc /dλ ij < 0 and dc /dq>0. he first inequality implies that as the rate of discharge of organic pollutants Q increases, the equilibrium level of dissolved oxygen C decreases. he second inequality suggests that C further decreases due to presence of various degradation processes. Also, it can be noted from the third inequality that C increases as the rate of input of dissolved oxygen q increases. 4. Stability analysis In our analysis, we assume that all the above equilibria exist. he local stability behavior of these equilibria has been studied and the result is stated in the following theorem: heorem. he equilibrium E i i =, 2 or 3 is unstable whenever E i+ exists. he equilibrium E 4 is locally asymptotically stable provided the following condition is satisfied: λ 0 K 2K 22 K 2 2 he proof is given in Appendix A. P > heorem 2. he equilibrium E 4 is nonlinearly stable in Ω provided the following two conditions are satisfied: λ 0 K 2K 22 P > 0, 4.2 K 2 2 α 0 λ 0 K 2K 22 λ K2 2 P K K 2 Q 2 K 2 + K > K 2 α 0 + K Q he proof is given in Appendix B. We note from the above that if λ 0 = 0, the above conditions are never satisfied. his shows that crowding coefficient of bacteria population density i.e. flaking off term stabilizes the system. he above theorems imply that the concentration of dissolved oxygen decreases as the discharge rate of organic pollutants increases. hey also show that if the cumulative rate of discharge of pollutants is very high, the concentration of dissolved oxygen may become negligible caused by various degradation processes. 5. A model for control of organic pollutants Here, we propose and analyze a nonlinear mathematical model to study the effect of control of discharge rate of organic pollutants on dissolved oxygen by using some effort 5. Let e be the effort applied to control the discharge of organic pollutants. It is assumed that the growth rate of effort applied to control the organic pollutants is proportional to the cumulative concentration of organic pollutants i.e. r, and that the rate of natural depletion of effort is proportional to e i.e. r 0 e. It is assumed further that the rate of decrease of organic pollutants in the water body is proportional to effort e i.e. γe. In the following model the other notations and assumptions for growth rates and depletion rates

9 988 A.K. Misra et al. / Nonlinear Analysis: Real World Applications of organic pollutants, bacteria, protozoa, lower order aquatic population and dissolved oxygen are the same as in model 2.. Keeping the above considerations in mind, the control dynamics of the system is governed by the following system of differential equations: d db dp dn dc = Q α 0 K B K 2 + K γ e, = λ K B K 2 + K α B λ 0 B 2 K 2 BP K 2 + K 22 B, = λ 2 K 2 BP K 2 + K 22 B α 2P λ 20 P 2 K 3 PN, = λ 3 K 3 PN α 3 N λ 30 N 2, = q α 4 C λ 34 K 3 PN λ α B λ 22 α 2 P λ 33 α 3 N λ 2 K B K 2 + K λ 23 K 2 BP K 2 + K 22 B, de = r r 0 e, 5. where 0>0, B0>0, P0>0, N0>0, C0>0, e0>0. In 5. γ is the control rate of cumulative density of organic pollutants, r is the growth rate of effort and r 0 is its depletion rate. As before we can find that the set Ω c, defined below, is a region of attraction for model 5.. { Ω c = 0 Q, 0 B R B, 0 P R P, 0 N R N, 0 C q, 0 e r } Q, 5.2 α 0 α 4 r 0 α 0 where all the initial values of the variables belong to set Ω c. 5.. Equilibrium analysis Model 5. has four equilibria in Ω c as described below the proof of existence of each of these equilibria is similar as discussed for model 2.. i Ẽ r0 Q r 0 α 0 +γr, 0, 0, 0, q r α 4, Q r 0 α 0 +γr. ii Ẽ 2 2, B 2, 0, 0, C 2, ẽ 2 exists provided the following are satisfied: and λ K K α Q K 2 α α 0 + γr /r 0 >0, 5.3 q λ 2 K 2 B 2 K 2 + K 2 λ α B 2 > 0, 5.4 where 2, B 2, C 2 and ẽ 2 are given by the following equations: Q α 0 + γr 2 K 2 B 2 = 0, r 0 K 2 + K 2 λ K 2 B 2 K 2 + K 2 α λ 0 B 2 = 0,

10 A.K. Misra et al. / Nonlinear Analysis: Real World Applications C 2 = K 2 B 2 q λ 2 λ α B 2, α 4 K 2 + K 2 ẽ = r r 0 2. iii Ẽ 3 3, B 3, P 3, 0, C 3, ẽ 3 exists provided the following conditions are satisfied: λ 20 λ K K α Q K 2 α α 0 + γr + K 2α 2 K Q + K 2 α 0 + γr > 0, 5.5 r 0 K 2 r 0 λ 0 + K 2 λ 20 K 2 + K 22 R B 2 λ 2 K 2 K 2 K 2 + K 22 R B K 22 λ2 K 2 R B α 2 > 0, 5.6 K 2 + K 22 R B λ 2 K 2 B 3 K 2 + K 22 B 3 α 2 > 0, 5.7 K 3 B 3 K 2 B 3 P 3 q λ 2 λ 23 λ α B 3 λ 22 α 2 P 3 > 0, 5.8 K 2 + K 3 K 2 + K 22 B 3 where 3, B 3, P 3, C 3 and ẽ 3 are given by the following equations: Q α 0 + γr 3 K 3 B 3 = 0, r 0 K 2 + K 3 λ K 3 K 2 + K 3 α = λ 0 B + P 3 = λ 2 K 2 B 3 α 2, λ 20 K 2 + K 22 B 3 K 2 λ 20 K 2 + K 22 B 3 λ 2 K 2 B 3 K 2 + K 22 B 3 α 2 C 3 = K 3 B 3 K 2 B 3 P 3 q λ 2 λ 23 λ α B 3 λ 22 α 2 P 3, α 4 K 2 + K 3 K 2 + K 22 B 3 ẽ 3 = r r 0 3., iv Ẽ 4, B, P,Ñ, C,ẽ exists, provided the following conditions are satisfied: λ K K α Q K 2 α α 0 + γr + K 2m K Q + K 2 α 0 + γr > 0, 5.9 r 0 K 2 l r 0 λ 0 + K 2 λ30 λ 2 K 2 K 2 K 2 + K 22 R B 2 l K 2 + K 22 R B K λ30 λ 2 K 2 R B 22 K 2 + K 22 R B m > 0, 5.0 λ 30 λ 2 K 2 B m>0, 5. K 2 + K 22 B

11 990 A.K. Misra et al. / Nonlinear Analysis: Real World Applications λ 3 K 3 P α 3 > 0, 5.2 q λ 2K B K 2 + K λ 23K 2 B P K 2 + K 22 B λ 34K 3 P Ñ λ α B λ 22 α 2 P λ 33 α 3 Ñ>0, 5.3 where l = λ 20 λ 30 + λ 3 K 2 3, m = λ 30α 2 K 3 α 3 and, B, P,Ñ, C,ẽ are given by the following equations: Q α 0 + γr r 0 K B K 2 + K λ K K 2 + K α = λ 0 B + P = l = 0, 5.4 K 2 lk 2 + K 22 B λ 30 λ 2 K 2 B K 2 + K 22 B m, 5.5 λ 30 λ 2 K 2 B K 2 + K 22 B m, 5.6 Ñ = λ 30 λ 3 K 3 P α 3, 5.7 C = q α 4 λ 2K B K 2 + K λ 23K 2 B P K 2 + K 22 B λ 34K 3 P Ñ λ α B λ 22 α 2 P λ 33 α 3 Ñ, 5.8 ẽ = r r In this case also it can be checked that as the rate of control of organic pollutants γ increases, the equilibrium level of the dissolved oxygen concentration C increases. As in the previous case, for model 5. also we can easily show that Ẽ i i =, 2or3 is unstable whenever Ẽ i+ exists. he local stability behavior of the equilibrium Ẽ 4, B, P,Ñ, C,ẽ is stated in the following theorem. heorem 3. he equilibrium Ẽ 4 is locally asymptotically stable if λ 0 K 2K 22 K 2 2 he proof is given in Appendix C. P> heorem 4. he equilibrium Ẽ 4 is nonlinearly stable in Ω c if the following conditions are satisfied: λ 0 K 2K 22 K 2 2 P>0, 5.2 α 0 λ 2 λ 0 K 2K 22 K K 2 Q P K2 2 > K 2 + K K 2 α 0 + K Q In this case also, we may note the stabilizing effect of λ 0. Further, these theorems imply that if an appropriate effort is applied to control organic pollutants, the equilibrium level of dissolved oxygen can be maintained at a desired level. he proof is given in Appendix D.

12 A.K. Misra et al. / Nonlinear Analysis: Real World Applications *, B * B Fig. 4. Global stability in B plane. 6. Numerical example o check the feasibility of our analysis regarding the existence of E 4 and corresponding stability conditions, we conduct some numerical computation by choosing the following values of the parameters in model 2.: Q = 3.0, α 0 =.0, K =.0, K 2 = 0., K =.0, λ =.0, α = 0.5, λ 0 = 0., K 2 =.0, K 2 = 2.0, K 22 =.0, λ 2 =.0, α 2 = 0.02, λ 20 = 0.25, K 3 =.0, λ 3 =.0, α 3 = 0.2, λ 30 = 0.25, q = 0.0, α 4 = 2.0, λ 2 = λ 23 = λ 34 = λ = λ 22 = λ 33 = It is found that under the above set of parameters, conditions for the existence of interior equilibrium E 4,B,P,N,C are satisfied and E 4 is given by = , B = , P = , N = , C = he eigenvalues of the Jacobian matrix M corresponding to this equilibrium E 4 are 2, i, i, i, i, which are either negative or have negative real parts. Hence E 4 is locally stable. It is pointed out here that for the above set of parameters, the conditions for local stability 4. and nonlinear stability 4.2 and 4.3 are also satisfied. Further, for the above set of parameters model 2. computer generated graphs of verses B and verses C are shown in Figs. 4 and 5, which indicates the global stability of,b in the B-plane and,c in C-plane. 7. Conclusion In this paper, we have proposed and analyzed a nonlinear mathematical model for the depletion of dissolved oxygen due to discharge of organic pollutants in a water body by considering biodegradation and biochemical processes in

13 992 A.K. Misra et al. / Nonlinear Analysis: Real World Applications C 4 *, C * Fig. 5. Global stability in C plane. the food chain involving bacteria, protozoa and an aquatic population for example a fish population in a water body. It has been shown that if the cumulative rate of discharge of pollutants increases, the equilibrium level of dissolved oxygen decreases, the amount of which depends on the rate of growth of oxygen as well as on various biodegradation and biochemical processes in the water body. It has also been shown that under certain conditions, if the cumulative rate of introduction of water pollutants is too high, the equilibrium concentration of the dissolved oxygen may become negligibly small, threatening the survival of biological species in a water body. A model to control the cumulative discharge of organic pollutants in a water body is also proposed and analyzed. It has been shown that the equilibrium concentration of the dissolved oxygen can be maintained at a desired level by using an appropriate effort to control the cumulative discharge of organic pollutants. Appendix A. Local stability analysis he general variational matrix M for system 2. is given as follows: K f, B K 2 + K λ K K 2 B K 2 B K M = 2 + K 2 f 2,B,P 0 0 K 2 + K 22 B λ 2 K 2 K 2 P, 0 K 2 + K 22 B 2 f 3 B,P,N K 3 P λ 3 K 3 N f 4 P, N 0 g, B g 2,B,P g 3 B, N g 4 P α 4 where f, B = α 0 + f 2,B,P= f 3 B,P,N= K K 2 B K 2 + K 2, λ K K 2 + K α 2λ 0 B K 2 K 2 P K 2 + K 22 B 2, λ 2 K 2 B K 2 + K 22 B α 2 2λ 20 P K 3 N, f 4 P, N = λ 3 K 3 P α 3 2λ 30 N,

14 g, B = g 3 B, N = A.K. Misra et al. / Nonlinear Analysis: Real World Applications λ 2K K 2 B K 2 + K 2, g 2,B,P= λ 2K K 2 + K + λ 23K 2 K 2 P K 2 + K 22 B 2 + λ α, λ 23K 2 B K 2 + K 22 B + λ 22α 2 + λ 34 K 3 N, g 4 P = λ 34 K 3 P + λ 33 α 3. A. We use eigenvalue method to study the local stability behavior of equilibria E i i =, 2, 3. Let M i be the matrix obtained from M after substituting for E i. For the equilibrium E, we note that one of the eigenvalues of M is λ K K α Q K 2 α 0 α /K 2 α 0 +K Q, which is positive whenever E 2 exists condition 3.. Hence, E is unstable whenever E 2 exists. For the equilibrium E 2 2,B 2, 0, 0,C 2, we note that one of the eigenvalues of M 2 is λ 2 K 2 B 2 /K 2+K 22 B 2 α 2, which is positive whenever E 3 exists as B 3 <B 2. hus, E 2 is unstable whenever E 3 exists. For the equilibrium E 3 3,B 3,P 3, 0,C 3, we note that one of the eigenvalues of M 3 is λ 3 K 3 P 3 α 3, which is positive whenever E 4 exists as P <P 3. his shows that E 3 is unstable whenever E 4 exists. Since in our study E 4 is the most interesting equilibrium from ecological point of view and its behavior cannot be described in a simple manner from M 4, we discuss its behavior by using Lyapunov s method. Proof of heorem. We linearize system 2. by using the transformations = + τ, B= B + b, P = P + p, N = N + n, C = C + c. A.2 Now we consider the following positive definite function: V = 2 τ 2 + m B b2 + m 2 P p2 + m 3 N n2 + m 4 c 2, A.3 and use the linearized system of 2. to get dv = f,b τ 2 m f 5 B,P b 2, m 2 λ 20 p 2 m 3 λ 30 n 2 m 4 α 4 c 2, m 4 g,b τc m 4 g 2,B,P bc m 4 g 3 B,N pc m 4 g 4 P nc, A.4 where the functions f and g i i =, 2, 3, 4 are given by A., and f 5 B, P = λ 0 K 2 K 22 P /K 2 + K 22 B 2. Now choosing m = K 2 + K /λ K 2, m 2 = m K 2 + K 22 B /λ 2 K 2, m 3 = m 2 /λ 3 we note that the coefficient of b 2 is negative under condition 4.. hus dv/ can be made negative definite by appropriately choosing a positive value for m 4 following usual methods of nonlinear analysis 4,7,6,7. Appendix B. Proof of heorem 2 o prove this theorem we consider the following positive definite function: V = 2 /2 + m B B B lnb/b + m 2 P P P lnp /P + m 3 N N N lnn/n + m 4 C C 2 /2. B.

15 994 A.K. Misra et al. / Nonlinear Analysis: Real World Applications Now using system 2. we get, dv K K 2 B 2 = K 2 + K K 2 + K m K 2 K 22 P K2 2 K 2 + K 22 BK 2 + K 22 B B B 2 α 0 2 K 2 K 22 m λ 0 K 2 + K 22 B 2 P B B 2 m 2 λ 20 P P 2 m 3 λ 30 N N 2 m 4 α 4 C C 2 + B B K K 2 K 2 + K K 2 + K m 4 C C λ 2 K K 2 B K 2 + K K 2 + K m 4 B B C C λ2 K K 2 + K + λ 23 K 2 K 2 P K 2 + K 22 BK 2 + K 22 B + λ α m 4 P P C C λ23 K 2 B K 2 + K 22 B + λ 34K 3 N + λ 22 α 2 m 4 N N C C λ 34 K 3 P + λ 33 α 3. Note that K2 2 K 2 + K 22 BK 2 + K 22 B > 0 inside the region of attraction Ω. Now choosing m = /λ,m 2 = m K 2 + K 22 B /λ 2 K 2, m 3 = m 2 /λ 3, we note that dv/ can be made negative definite inside Ω if λ 0 K 2K 22 P > 0, K 2 2 α 0 λ 0 K 2K 22 λ K2 2 P K K 2 Q 2 K 2 + K > 0 B.4 K 2 α 0 + K Q by making an appropriate choice for m 4 as pointed out earlier. B.2 B.3 Appendix C. Proof of heorem 3 We linearize system 5. by using the following transformations: = + τ, B = B + b, P = P + p, N = Ñ + n, C = C + c, e =ẽ + e. Now using the following positive definite function: V = τ 2 + m 2 B b2 + m 2 P p2 + m 3 Ñ n2 + m 4 c 2 + m 5 e 2, C. and use the linearized system of 5. to get dv = f, Bτ 2 m f 5 B, Pb 2 m 2 λ 20 p 2 m 3 λ 30 n 2 m 4 α 4 c 2 γr 0 e 2 r m 4 g, Bτc m 4 g 2, B, Pbc m 4 g 3 B,Ñpc m 4 g 4 P nc and m s chosen as positive constants in the following manner: m = K 2 + K /λ K, m 2 = m K 2 + K 22 B/λ 2 K 2, m 3 = m 2 /λ 3,m 5 = γ/r. C.2

16 If the following condition is satisfied: λ 0 K 2K 22 K 2 2 P>0. A.K. Misra et al. / Nonlinear Analysis: Real World Applications then dv/ can be made negative definite by appropriate choice of m 4 > 0. Hence Ẽ 4 is locally stable under above condition. Appendix D. Proof of heorem 4 We prove this theorem by using the following positive definite function: V = m B B B ln B B + m 2 P P P ln P P + m 3 N Ñ Ñ ln Ñ + N 2 m 4C C m 5e ẽ 2, D. where m,m 2,m 3,m 4,m 5 are some positive constants to be chosen appropriately. Using system 5. with the constants m,m 2,m 3 and m 5 chosen as follows: m = K 2 + K /λ K, m 2 = m K 2 + K 22 B/λ 2 K 2, m 3 = m 2 /λ 3,m 5 = γ/r,weget dv K K 2 B 2 = K 2 + K K 2 + K m K 2 K 22 P K2 2 B B 2 m 5 r 0 e ẽ 2 K 2 + K 22 BK 2 + K 22 B α 0 2 m λ 0 K 2K 22 P K2 2 B B 2 m 2 λ 20 P P 2 m 3 λ 30 N Ñ 2 m 4 α 4 C C 2 K K 2 + B B K 2 + K K 2 + K λ 2 K K 2 B m 4 C C K 2 + K K 2 + K λ 2 K m 4 B BC C K 2 + K + λ 23 K 2 K 2 P K 2 + K 22 BK 2 + K 22 B + λ α m 4 P P λ 23 K 2 B C C K 2 + K 22 B + λ 34K 3 Ñ + λ 22 α 2 m 4 N ÑC Cλ 34 K 3 P + λ 33 α 3. Now dv/ can be made negative definite in Ω c provided the following conditions as given in the theorem are satisfied: λ 0 K 2K 2 K 2 2 P>0 and α 0 λ 0 K 2K 22 K K 2 P λ K2 2 K 2 + K for a suitable choice of m 4 > 0. 2 Q > 0 K 2 α 0 + K Q D.2

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