Dynamics of a prey-generalized predator system with disease in prey and gestation delay for predator

Model. Earth Syst. Environ. (2016) 2:52 DOI 10.1007/s40808-016-0096-8 ORIGINAL ARTICLE Dynamics of a prey-generalized predator system with disease in prey and gestation delay for predator Harkaran Singh1,2 • Joydip Dhar3 • Harbax S. Bhatti4 Received: 29 December 2015 / Accepted: 18 February 2016 / Published online: 22 March 2016 Ó Springer International Publishing Switzerland 2016 Abstract In the present study, a prey-generalized predator model is proposed with disease in the prey and gestation delay for predator. The asymptotic behavior of the model is studied for all the feasible equilibrium states. The criterion for local stability of the system are established around steady states and thresholds for Hopf bifurcation are determined at the endemic as well as disease-free state. The respective sensitive indices of the variables are identified at the endemic state by performing the sensitivity analysis. Further numerical simulations have been carried out to justify our analytic findings. Keywords Prey-predator model  Disease in prey  Gestation delay  Hopf bifurcation  Sensitivity analysis & Harkaran Singh Joydip Dhar Harbax S. Bhatti 1 IKG-Punjab Technical University, Kapurthala 144601, Punjab, India 2 Department of Applied Sciences, Khalsa College of Engineering and Technology, Amritsar 143001, Punjab, India 3 Department of Applied Sciences, ABV-Indian Institute of Information Technology and Management, Gwalior 474015, MP, India 4 Department of Applied Sciences, B. B. S. B. Engineering College, Fatehgarh Sahib 140406, Punjab, India Introduction The interactions between prey and predator living in the same environment is a fascinating field in the bio-mathematical literature starting with the pioneer work of Lotka (1925) and Volterra (1926). Many mathematicians and ecologists studied the dynamical behavior of the preypredator system in ecology and contributed to the growth of the population models (Dhar and Jatav 2013; Dubey 2007; Freedman 1980; Jeschke et al. 2002; Kooij and Zegeling 1996; Ma and Takeuchi 1998; Singh et al. 2015; Dhar et al. 2015; Sen et al. 2012; Murray 2002; Robinson 1998; Tripathi et al. 2015). Further, the correlation between the disease and the prey-predator system is a topic of significant interest, and the fusion of ecology and epidemiology is a comparatively new branch of study, known as eco-epidemiology. It is a well known fact that the predator is more vulnerable to the infected prey because the infected prey may become weaker and less active so that they may be easily caught by the predator, and the same concept was modeled by various researchers (Moore et al. 2002; Hethcote et al. 2004; Liu and Wang 2010; Hadeler and Freedman 1989). But, there is also a possibility that, the predator gets infected due to consumption of the infected prey and dies out more rapidly. In this latter case the growth of the predator will depend on the healthy prey. Further, there will be the lack of the healthy prey due to disease in the prey population and therefore, the predator depends on the alternative food for their survival. Also, in population dynamics, growth is not instantaneous, it will take some time to perform, for example, the predator populations take some time to born a new offspring after mating is known as gestation delay. The simplest preypredator models cannot capture the rich variety of dynamics and the inclusion of the gestation delay in these 123 52 Page 2 of 9 Model. Earth Syst. Environ. (2016) 2:52 models makes them more realistic (Driver 1977; Beretta et al. 1995; Brauer 1990; Jin and Ma 2006). In this paper, we have analyzed a prey-predator model with gestation delay for predator growth. We have considered that prey population is suffered from a communicable disease and the predator depends on alternative resources for their survival. This paper is organized as follows: in Sect. 2, formulation of the mathematical model is presented. In Sect. 3, positivity and boundedness of the system has been obtained. In Sect. 4, the stability criterion of the system is discussed at all the feasible equilibrium states and obtained the conditions for the existence of Hopf bifurcation at the disease-free and endemic equilibrium states. In Sect. 5, the sensitive parameters of the state variables are identified and in Sect. 6, we presented numerical simulations in support of our analytical findings. Finally, the results has been concluded in the last section. Formulation of mathematical model The assumptions of the proposed model are: (i) (ii) (iii) (iv) (v) In a particular habitat, there are two populations; prey and predator. The prey population is suffered from a communicable disease, and it is divided into two mutually exclusive classes, susceptible S and infective I at any time t. The density of predator population at any time t is P. We suppose that due to the disease in the prey population, the infected individuals are unable to produce offsprings. The predator might get infected due to consumption of infected prey and dies out with a fixed rate h. s is a gestation delay in predator growth. The predator depends on healthy prey for their growth and due to lack of healthy prey, the predator also depends on alternative resources. The proposed system is of the form:   dS SþI bSP ¼ aS 1   bSI;  dt k Sþl ð1Þ dI ¼ bSI  b0 IP  d0 I; dt ð2Þ dP mbSðt  sÞPðt  sÞ ¼ cP  hIP þ  dP2 ; dt Sðt  sÞ þ l ð3Þ with initial conditions: SðaÞ ¼ w1 ðaÞ; IðaÞ ¼ w2 ðaÞ; PðaÞ ¼ w3 ðaÞ; w1 ð0Þ [ 0; w2 ð0Þ [ 0; w3 ð0Þ [ 0; 123 where a 2 ½s; 0 and w1 ðaÞ, w2 ðaÞ, w3 ðaÞ 2 Cð½s; 0; R3þ Þ, the Banach space of continuous functions mapping the interval ½s; 0 into R3þ , where R3þ ¼ fðx1 ; x2 ; x3 Þ : xi  0; i ¼ 1; 2; 3g. The detail description of the parameters is stated in Table 1. Positivity and boundedness of the system We state and prove the following lemmas for the positivity and boundedness of the solution of the system (1–3): Lemma 1 The solution of the Eqs. (1–3) with initial conditions are positive, for all t  0. For t 2 ½0; s, the Eq. (1) can be rewritten as   dS SþI bSP   aS  bSI;  dt k Sþl Proof and it follows that n R   o t bPS exp  0 aIk þ SðSþlÞ þ bI du n R   o [ 0: SðtÞ  Rt t bPS Sð0Þ þ 0 a exp  0 aIk þ SðSþlÞ þ bI du dv For t 2 ½0; s, the Eq. (2) can be rewritten as dI   b0 IP  d0 I; dt which evidences that  Z t  IðtÞ  Ið0Þ exp  ðd0 þ b0 PÞdu [ 0: 0 The Eq. (3) for t 2 ½0; s can be rewritten as dP   hIP  dP2 ; dt which implies that Rt expf 0 hIdug Rt [ 0: Rt PðtÞ  Pð0Þ þ 0 d expf 0 hIdug Similarly, for the intervals ½s; 2s; ::::; ½ns; ðn þ 1Þs; n 2 N, it can be proved that S(t), I(t) and P(t) are positive. Thus by induction, S(t), I(t) and P(t) are positive for all t  0. h Lemma 2 The solution of the Eqs. (1–3) with initial conditions is uniformly bounded in X, where   k2 X ¼ ðS; I; PÞ : 0  SðtÞ þ IðtÞ þ PðtÞ  ; k1 2 k1 ¼ minfd1 ; d2 ; d3 g and k2 ¼ ak þ cd . Proof Le (...truncated)