The effect of an external toxicant on a biological species in case of deformity: a model

Modeling Earth Systems and Environment, Aug 2016

In this paper, a mathematical model is proposed and analyzed to study the effect of an external toxicant on a biological species. Here, we have considered that the toxicant is constantly emitted in the environment form some external source and after-effect of this external toxicant some members of biological species shows deformity as incapable in reproduction. The analytical results of model system are established by stability analysis and Hopf-bifurcation theory. The model’s results show, when emission of external toxicant increases, total population density decreases and density of deformed subclass increases. For highly emission of external toxicant, system become unstable and shows a supercritical Hopf-bifurcation. To verify the analytical results, a numerical simulation is provided.

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The effect of an external toxicant on a biological species in case of deformity: a model

Model. Earth Syst. Environ. The effect of an external toxicant on a biological species in case of deformity: a model Anuj Kumar Agarwal 0 1 2 3 4 A. W. Khan 0 1 2 3 4 A. K. Agrawal 0 1 2 3 4 0 Department of Mathematics, Integral University , Lucknow, Uttar Pradesh , India 1 A. K. Agrawal 2 Department of Mathematics, Amity University , Lucknow, Uttar Pradesh , India 3 Mathematics Subject Classification 34 C60 92D25 93A30 4 & Anuj Kumar Agarwal In this paper, a mathematical model is proposed and analyzed to study the effect of an external toxicant on a biological species. Here, we have considered that the toxicant is constantly emitted in the environment form some external source and after-effect of this external toxicant some members of biological species shows deformity as incapable in reproduction. The analytical results of model system are established by stability analysis and Hopf-bifurcation theory. The model's results show, when emission of external toxicant increases, total population density decreases and density of deformed subclass increases. For highly emission of external toxicant, system become unstable and shows a supercritical Hopf-bifurcation. To verify the analytical results, a numerical simulation is provided. Mathematical model; Biological species; Toxicant; Deformity; Hopf-bifurcation - 37L10 Introduction Mathematical models are used in large-scale to predict the various nature of real life problems; e.g. in ecology, epidemiology, ecotoxicology and other many problems. Many researchers also used mathematical models to predict the growth of biological species in toxic environment. They have proposed and analyzed mathematical models by considering different cases, such as effect of a single toxicant or more than one toxicant on biological species, allelopathy case, deformity in a subclass of species, etc. (Freedman and Shukla 1991; Shukla and Agrawal 1999; Shukla et al. 2003; Agrawal and Shukla 2012; Kumar et al. 2016) , to provide important insights for the effect of toxicants on biological species. In particular, Freedman and Shukla (1991) have studied the effect a single toxicant on a species with a consideration that toxicant affected on the growth rate and decreasing the carrying capacity of the environment. Shukla and Agrawal (1999) have proposed a model by considering a situation in which toxicant emitted by a biological species and decreased the density of other biological species (case of allelopathy). As an interesting observable fact, Agrawal and Shukla (2012) studied a model for the after-effect of a single toxicant a subclass of biological species shows deformity as incapable in reproduction. Here, it is assumed that toxicant is emitted in the environment from some external source. Further, understand the case of deformity after-effect of a toxicant in more meaningful manner, Kumar et al. (2016) proposed and analyzed a model with an assumption that toxicant emitted by biological species itself. They have shown, toxicant decreased the total population density and a subclass of species suffers from deformity. For higher emission rate, the model system becomes unstable. In this paper, we proposed and analyzed a mathematical model to study the effect of an external toxicant on a biological species a subclass of which is severely affected and gets deformed. This case is similar to the case studied by Agrawal and Shukla (2012) . But, the proposed model in this study is a modified version of the model by Agrawal and Shukla (2012) . The results obtained by modified model are more closer to real life in case of deformity. In this study, we also check the existence of hopf-bifurcation and the nature of bifurcating periodic solutions. Mathematical model We assume a biological species of population density N(t) at time t, is logistically growing and surviving in a polluted environment. This polluted environment having a toxicant which is constantly emitted in the environment from some external sources. The environmental concentration of this toxicant is T(t) at time t. We assume that U(t) is the concentration of toxicant T(t), taken up by the biological species N(t) at time t. This toxicant is decreasing the growth rate of species N(t) as well as a subclass of species with population density ND?t? shows deformity as incapability in reproduction. The remaining population density which is free from deformity is assumed as NA?t?. Keeping these facts in mind, we propose the following mathematical model: ddNtA ? ?b dND dt ? r1UNA dT dt ? Q dU dt ? cTN d?NA r1UNA rNDN K?T ? rNAN K?T? All the parameters considered in the model are positive constants. b and d are the natural birth and death rate of biological species. r represents the intrinsic growth rate of biological species. Q is the rate at which external toxicant is constantly emitted in the environment. The external toxicant T(t) is uptaken by the species at the rate c. Aftereffect of this toxicant, the deformed-free population density decreases at the rate r1. a is the mortality rate of deformed population due to high toxicity in the environment. d and b are naturally depletion rates of T(t) and U(t) respectively. U(t) is depleted at the rate m due to die out of some members of species and a fraction p of this depletion is reentered into the environment. c [ 0 is a proportionality constant used to calculate the initial uptake concentration of toxicant. K(T) is a decreasing function of T to measure the carrying capacity of the environment. i.e. K?0? ? K0 [ 0; K?T? [ 0; \0; for T [ 0 dK dT where K0 is the carrying capacity of the toxic free environment. To make the model system (1) free from NA?t?, we reduce it into following model system (3) by using the fact that N?t? ? NA?t? ? ND?t?. dN dt ? rN dT dt ? Q dU dt ? cTN rN2 K?T ? ND? ?a ? b?ND rNDN K?T? dT cTN ? pmNU bU mNU N ? ?r r1U?K?T ? r r1UNK?T? ND ? rN ? ?r1U ? a ? d?K?T? Q?b ? mN? T ? f ?N? QcN U ? f ?N? ? h?N? ?say? ? g?N? ?say? where f ?N? ? db ? ?cb ? dm?N ? cm?1 p?N2 ?4e? The Eq. (4c) shows that T is directly proportional to the parameter Q and from the Eq. (2), carrying capacity K(T) decreases as T increase. Hence, the carrying capacity of the environment decreases when the emission rate of external toxicant Q increases. Let p?N ? m2?1 p?N2g\0 F?N? ? rN at N ? 0, F?0? ? rK at N ? K0 ?r Q d r1h?N??K?g?N?? \0 F?K0? ? rK0 ?r r1h?K0??K?g?K0?? [ 0 Eqs. (6) and (7) show that F?N? ? 0 has a solution in the interval ?0; K0 . Also, The root N of F?N? ? 0 is unique, if ddNF ? r ? r1K?g?N?? ddNh ?r r1h?N?? ddKT ddNg [ 0 ?5? ?6? ?7? ?8? ?9a? ?9b? ?10? E2. X_ ? M2X ? N2 where, 2 n 3 X ? 6646 nsd 7577; u Moreover, m11 ? r m21 ? r1U from, Eqs. (4c) and (4d) dg dN ? Qc 2 f 2?N? fb ? 2bm?1 dh Qc dN ? f 2?N? fdb cm?1 p?N2g since, ddKT \0 (from (2)) and ddNg (from (9a)) so, ?r dK dg r1h?N?? dT dN The equation F?N? ? 0 has a unique root N , only when dh r ? r1K?g?N?? dN here, n; nd; s and u are taken as small perturbations around So, the model system (3) can be written in the terms of n; nd; s and u as follows: ?11? In the Eq. (11), M2X and N2 are showing the linear and non-linear parts of the model system (3) respectively and M2 is a Jacobian matrix corresponding to the equilibrium point E2. Thus, the characteristic equation of M2 can be written as: p?x? ? x4 ? c1x3 ? c2x2 ? c3x ? c4 ?cj Q?Q [ 0 for ?j ? 1; . . .; 4? ?H2 Q?Q ? ?c1c2 Applying the Routh?Hurwitz Criterion on the characteristic Eq. (12), all the eigenvalues of Jacobian matrix M2 are either negative or having negative real parts iff Hence, we can state the following theorem to set up the local asymptotically stablility corresponding to the equilibrium point E2. Theorem 1 The equilibrium point E2 of model system (3) is locally asymptotically stable under the conditions (13). Existence of Hopf-bifurcation The model system (3), has a possibility of Hopf-bifurcation (Hassard et al. 1981; Kuznetsov 2004; Seydel 2009) corresponding to the equilibrium point E2. By treating Q (i.e. the emission rate of external toxicant) as a bifurcation parameter, we check the existence of Hopf-bifurcation. It is obvious that a Hopf-bifurcation may exist if all the eigenvalues of Jacobian matrix are having negative real parts except a purely imaginary complex conjugate pair. In this case, the Jacobian matrix M2 having four eigenvalues xj ? Rj ? iIj ?j ? 1; . . .; 4? (say). So, the Hopf-bifurcation exist only when R1; R2 ? 0; I1 ? I2 ?6 0 & R3; R4\0 at the critical value Q ? Q (say). According to the Liu?s criterion (Liu 1994) , the model system (3) undergoes a Hopf-bifurcation at the critical value Q ? Q [ 0, if ?14a? ?14b? ?14c? ?14d? dRj dQ Q?Q 6?0; for j ? 1; 2 A New Detecting Method For Conditions of Existence of Hopf-bifurcation (Jiaqi and Zhujun 1995) describe the last condition (14d) in the terms of coefficients of characteristic Eq. (12) as follows: dR dQ Q?Q ? "ddQ ?c1c2c3 2c1?4c4 20 v 0 0 3 6 v 0 0 J ? P 1M2P ? 660 0 J1 00 7577 and 4 0 0 0 J2 2F1?y1;y2;y3;y4?3 f ? 6664ff23??yy11;;yy22;;yy33;;yy44??7757 f4?y1;y2;y3;y4? Here, 4 4 4 4 n ? XP1jyj; nd ? XP2jyj; s ? XP3jyj; u ? XP4jyj j?1 j?1 j?1 j?1 Now, we evaluated the following quantities at critical value of parameter Q ? Q and ?y1;y2;y3;y4? ? ?0;0;0;0?. i C1?0? ? 2v g20g11 2jg11j2 Hence, the following theorem express the nature of bifurcating periodic solutions. Theorem 3 If l2 [ 0 (or l2\0), the model system (3) shows a supercritical (or subcritical) Hopf-bifurcation and the bifurcating periodic solutions exist for Q [ Q (or Q\Q ), if b2\0 (or b2 [ 0), the bifurcating periodic solutions are stable (or unstable), if s2 [ 0 (or s2\0), the period of bifurcating solutions increases (or decreases). Numerical simulation We provide numerical simulation to back up our analytical results for the model system (3). A matlab package MATCONT (Dhooge et al. 2003) is used for the graphical representation of model system (3). We assume, the carrying capacity function as b1T 1 ? b2T K?T ? ? K0 and a set of parameters as: b ? 0:55; a ? 0:0002; p ? 0:02; b1 ? 0:02; d ? 0:0006; d ? 0:08; m ? 0:0002; b2 ? 1:0 Fig. 1 Time-series graph of total and deformed population corresponding to the parameter Q ?18? ?19a? ?19b? The equilibrium point E2 contains the value N ? 9:5989; ND ? 0:3834; T The condition (10) holds and ddNF ? 0:5495 [ 0, which show that N is unique, in addition E2 is unique. The local stability conditions (13) corresponding to E2 are also satisfied. Figure 1 shows the total density and density of deformed subclass of biological species corresponding to the parameter Q (the remaining parameters are same as (19b)). The Fig. 1 shows that when the emission rate of external toxicant increases, the total density N decrease and the density of deformed subclass ND firstly increase then decrease with N. For large emission rate Q?? 0:950? both densities are oscillating. Figure 2 shows the real and imaginary parts of eigenvalues of Jacobian matrix M2 corresponding to the parameter Q. The real parts of all eigenvalues (i.e. Ri\0, i ? 1; ; 4) are negative for Q\Q ?? 0:83648?. At Q ? Q two eigenvalues become purely imaginary (i.e. R1 ? R2 ? 0 and I1 ? I2 6? 0), which confirms that a model system (3) undergoes a Hopf-bifurcation at Q ? 0:83648. Figure 3 shows the densities of both populations N and ND with respect to the emission rate of external toxicant Q. Both densities N and ND become stable at equilibrium level for Q\Q . After crossing the critical value Q ?? 0:83648?, the equilibrium point losses its stability and a supercritical Hopf-bifurcation occurs (since l2 ? 2:2456 10 4 [ 0). Both densities start oscillating around their equilibrium level with stable bifurcating periodic orbits (since b2 ? 3:5116 10 7\0). Conclusion A mathematical model is proposed to examine the growth of biological species in the case a subclass of species shows deformity, when an external toxicant is constantly emitted in the environment. The analytical results of model show, as emission rate of external toxicant increases, total population density decreases and density of deformed subclass population firstly increase then decrease with total population density. If emission rate crosses the critical value, the model system shows a supercritical Hopf-bifurcation and all the bifurcating periodic solutions of model system are stable. Appendix A: Proof of the region of attraction X Proof From the model system (3), we have dN dt rN rN2 K0 ? r 1 Thus, lim supt!1 N?t? Also, we have N K0 K0 N dT dU dt ? dt ? Q Q dT bU dm?T ? U? ?1 p?mNU where dm ? min?d; b?: Thus, lim supt!1?T?t? ? U?t?? dQm From the second equation of model (3), we have ?a ? d?ND Q r1 dm ?K0 ND? Thus, lim supt!1 ND?t? r1QK0 ?r1Q?dm?a?d? providing the region of attraction X: h Agrawal AK , Shukla JB ( 2012 ) Effect of a toxicant on a biological population causing severe symptoms on a subclass . S Pac J Pure Appl Math 1 ( 1 ): 12 - 27 Dhooge A , Govaerts W , Kuznetsov YA ( 2003 ) MATCONT a MATLAB package for numerical bifurcation analysis of ODEs . ACM Trans Math Softw (TOMS) 29 : 141 - 164 Freedman HI , Shukla JB ( 1991 ) Models for the effect of toxicant in single species and predator-prey systems . J Math Biol 30 : 15 - 30 Hassard B , Kazarinoff N , Wan Y ( 1981 ) Theory and application of Hopf bifurcation . London mathematical society lecture note series . Cambridge University Press, Cambridge Jiaqi S , Zhujun J ( 1995 ) A new detecting method for conditions of existence of hopf bifurcation . Acta Math Appl Sin 11 ( 1 ): 79 - 93 Kumar A , Agrawal AK , Hasan A , Misra AK ( 2016 ) Modeling the effect of toxicant on the deformity in a subclass of a biological species . Model Earth Syst Environ 2 : 1 - 4 Kuznetsov Y ( 2004 ) Elements of applied bifurcation theory . Springer, NewYork Liu W ( 1994 ) Criterion of Hopf bifurcation without using eigenvalues . J Math Anal Appl 182 : 250 - 256 Seydel R ( 2009 ) Practical bifurcation and stability analysis , vol 5 , 3rd edn. Springer, Berlin, p 11 Shukla JB , Agrawal AK ( 1999 ) Some mathematical models in ecotoxicology; effects of toxicants on biological species . Sadhana 24 : 25 - 40 Shukla JB , Agrawal AK , Sinha P , Dubey B ( 2003 ) Modeling effects of primary and secondary toxicants on renewable resources . Nat Resour Model 16 : 99 - 120


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Anuj Kumar Agarwal, A. W. Khan, A. K. Agrawal. The effect of an external toxicant on a biological species in case of deformity: a model, Modeling Earth Systems and Environment, 2016, 148, DOI: 10.1007/s40808-016-0203-x