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 & Anuj Kumar Agarwal 1 Mathematics Subject Classification 34 C60 92D25 93A30 2 Department of Mathematics, Amity University , Lucknow, Uttar Pradesh , India 3 Department of Mathematics, Integral University , Lucknow, Uttar Pradesh , India 4 A. K. Agrawal 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 NDðtÞ shows deformity as incapability in reproduction. The remaining population density which is free from deformity is assumed as NAðtÞ. Keeping these facts in mind, we propose the following mathematical model: ddNtA ¼ ðb dND dt ¼ r1UNA dT dt ¼ Q dU dt ¼ cTN dÞNA r1UNA rNDN KðT Þ rNAN KðTÞ 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 ð2Þ ð3Þ ð4aÞ ð4bÞ ð4cÞ ð4dÞ The region of attraction X that attracts all the positive solution of model system (3), is as follows: X ¼ ðN; ND; T; UÞ : 0 N K0; 0 ND r1QK0 ðr1Q þ dmða þ dÞÞ ; 0 T þ U Q dm where dm ¼ minðd; bÞ. The proof of region of attraction X is given in Appendix Equilibrium points The model system (3) have two non-negative equilibrium points E1ð0; 0; Qd ; 0Þ and E2ðN ; ND; T ; U Þ. Here, the existence of E1 is understandable, hence omitted. The existence of E2 is as follows: The values N ; ND; T and U of the equilibrium point E2 are the positive solutions of the following non-linear equations, 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. ð1Þ N ¼ ðr r1UÞKðT Þ r r1UNKðTÞ ND ¼ rN þ ðr1U þ a þ dÞKðTÞ Qðb þ mNÞ T ¼ f ðNÞ QcN U ¼ f ðNÞ ¼ hðNÞ ðsayÞ ¼ gðNÞ ðsayÞ where f ðNÞ ¼ db þ ðcb þ dmÞN þ cmð1 pÞN2 ð4eÞ 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 pÞN þ m2ð1 pÞN2g\0 FðNÞ ¼ rN at N ¼ 0, Fð0Þ ¼ rK at N ¼ K0 ðr Q d r1hðNÞÞKðgðNÞÞ \0 FðK0Þ ¼ rK0 ðr r1hðK0ÞÞKðgðK0ÞÞ [ 0 Eqs. (6) and (7) show that FðNÞ ¼ 0 has a solution in the interval ½0; K0 . Also, The root N of FðNÞ ¼ 0 is unique, if ddNF ¼ r þ r1KðgðNÞÞ ddNh ðr r1hðNÞÞ ddKT ddNg [ 0 ð5Þ ð6Þ ð7Þ ð8Þ ð9aÞ ð9bÞ ð10Þ E2. X_ ¼ M2X þ N2 where, 2 n 3 X ¼ 6646 nsd 7577; u Moreover, m11 ¼ r m21 ¼ r1U from, Eqs. (4c) and (4d) dg dN ¼ Qc 2 f 2ðNÞ fb þ 2bmð1 dh Qc dN ¼ f 2ðNÞ fdb cmð1 pÞN2g since, ddKT \0 (from (2)) and ddNg (from (9a)) so, ðr dK dg r1hðNÞÞ dT dN The equation FðNÞ ¼ 0 has a unique root N , only when dh r þ r1KðgðNÞÞ dN Dynamical behavior corresponding to E1 The Jacobian matrix M1 corresponding to the equilibrium points E1 is as follows 2 r ða þ bÞ 0 0 3 M1 ¼ 6666664 c0QcdQ ða00þ dÞ 00d 00b 7777757 d Since, one of the eigenvalue of Jacobian matrix M1 is r [ 0 and all the other eigenvalues are ða þ dÞ; d; b\0, which confirm that E1 is a saddle point locally unstable manifold in N direction and locally stable manifold in ND T U space. Dynamical behavior corresponding to E2 To study the dynamical behavior, we linearize the model system (3) corresponding to the equilibrium point E2 ¼ ðN ; ND; T ; U Þ by taking the following transformation: N ¼ N þ n; ND ¼ ND þ nd; T ¼ T þ s; U ¼ U þ u: 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: ð11Þ M2 is a Jacobian matrix corresponding to the equilibrium point E2. Thus, the characteristic equation of M2 can be written as: pðxÞ ¼ x4 þ c1x3 þ c2x2 þ c3x þ c4 ½cj Q¼Q [ 0 for ðj ¼ 1; . . .; 4Þ ½H2 Q¼Q ¼ ½c1c2 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 ¼ Rj þ iIj ðj ¼ 1; . . .; 4Þ (say). So, the Hopf-bifurcation exist only when R1; R2 ¼ 0; I1 ¼ I2 ¼6 0 & R3; R4\0 at the critical value Q ¼ Q (say). According to the Liu’s criterion (Liu 1994) , the model system (3) undergoes a Hopf-bifurcation at the critical value Q ¼ Q [ 0, if ð14aÞ ð14bÞ ð14cÞ ð14dÞ dRj dQ Q¼Q 6¼0; for j ¼ 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 Q¼Q ¼ "ddQ ðc1c2c3 2c1ð4c4 20 v 0 0 3 6 v 0 0 J ¼ P 1M2P ¼ 660 0 J1 00 7577 and 4 0 0 0 J2 2F1ðy1;y2;y3;y4Þ3 f ¼ 6664ff23ððyy11;;yy22;;yy33;;yy44ÞÞ7757 f4ðy1;y2;y3;y4Þ Here, 4 4 4 4 n ¼ XP1jyj; nd ¼ XP2jyj; s ¼ XP3jyj; u ¼ XP4jyj j¼1 j¼1 j¼1 j¼1 Now, we evaluated the following quantities at critical value of parameter Q ¼ Q and ðy1;y2;y3;y4Þ ¼ ð0;0;0;0Þ. l2 ¼ 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 þ b2T KðT Þ ¼ K0 and a set of parameters as: b ¼ 0:55; a ¼ 0:0002; p ¼ 0:02; b1 ¼ 0:02; d ¼ 0:0006; d ¼ 0:08; m ¼ 0:0002; b2 ¼ 1:0 Fig. 1 Time-series graph of total and deformed population corresponding to the parameter Q ð18Þ ð19aÞ ð19bÞ The equilibrium point E2 contains the value N ¼ 9:5989; ND ¼ 0:3834; T The condition (10) holds and ddNF ¼ 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 Qð¼ 0:950Þ 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 ¼ 1; ; 4) are negative for Q\Q ð¼ 0:83648Þ. At Q ¼ Q two eigenvalues become purely imaginary (i.e. R1 ¼ R2 ¼ 0 and I1 ¼ I2 6¼ 0), which confirms that a model system (3) undergoes a Hopf-bifurcation at Q ¼ 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 ð¼ 0:83648Þ, the equilibrium point losses its stability and a supercritical Hopf-bifurcation occurs (since l2 ¼ 2:2456 10 4 [ 0). Both densities start oscillating around their equilibrium level with stable bifurcating periodic orbits (since b2 ¼ 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 ¼ r 1 Thus, lim supt!1 NðtÞ Also, we have N K0 K0 N dT dU dt þ dt ¼ Q Q dT bU dmðT þ UÞ ð1 pÞmNU where dm ¼ minðd; bÞ: Thus, lim supt!1ðTðtÞ þ UðtÞÞ dQm From the second equation of model (3), we have ða þ dÞND Q r1 dm ðK0 NDÞ Thus, lim supt!1 NDðtÞ r1QK0 ½r1QþdmðaþdÞ 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 . 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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