Abstract
In this paper, a pest-natural enemy dynamics with maturation delay for pest species is proposed. Here, pest is categorized into two stages, namely immature and mature and the natural enemy only harvest mature pest population with Holling type-II interaction. The boundedness and positivity of the solution of the system are presented. The asymptotic behavior of the system is studied and analyzed for all feasible equilibrium points. Sensitivity analysis of the system at interior equilibrium point for the system parameters is performed, and respective sensitive indices of the variables are identified. Finally, numerical simulations are presented to support our analytic results.
Similar content being viewed by others
References
Thomas, M.B., & Willis, A.J. (1998). Biocontrol-risky but necessarys. Trends in Ecology and Evolution, 13, 325–329.
Parrella, M.P., Heinz, K.M., Nunney, L. (1992). Biological control through augmentative releases of natural enemies: a strategy whose time has come. American Entomologist, 38(3), 172–179.
Kishimba, M.A., Henry, L., Mwevura, H., Mmochi, A.J., Mihale, M., Hellar, H. (2004). The status of pesticide pollution in Tanzania. Talanta, 64(1), 48–53.
Weaver, R.D., Evans, D.J., Luloff, A.E. (1992). Pesticide use in tomato production: consumer concerns and willingness-to-pay. Agribusiness, 8(2), 131–142.
Ang, L, Dongfang, X., Song, Y. (2016). https://doi.org/10.12783/dtetr/iect2016/3715.
Aiello, W.G., & Freedman, H.I. (1990). A time-delay model of single-species growth with stage-structure. Mathematical Biosciences, 101(2), 139–153.
Song, Y., & Peng, Y. (2006). Stability and bifurcation analysis on a logistic model with discrete and distributed delays. Applied Mathematics and Computation, 181(2), 1745–1757.
Song, Y., & Wei, J. (2005). Local Hopf bifurcation and global periodic solutions in a delayed predator-prey system. Journal of Mathematical Analysis and Applications, 301(1), 1–21.
Wang, W., Mulone, G., Salemi, F., Salone, V. (2001). Permanence and stability of a stage-structured predator-prey model. Journal of Mathematical Analysis and Applications, 262(2), 499–528.
Arino, O., Hbid, M.L., Dads, E.A. (2006). Delay differential equations and applications, 205, by Springer, P.O. Box 17, 3300 AADordrecht, The Netherlands.
Cui, J., & Chen, L.S. (2001). Permanence and extinction in logistic and Lotka–Volterra systems with diffusion. Journal of Mathematical Analysis and Applications, 258(2), 512–535.
Faria, T. (2001). Stability and bifurcation for a delayed predator–prey model and the effect of diffusion. Journal of Mathematical Analysis and Applications, 254(2), 433–463.
Jiao, J.J., Chen, L.S., Cai, S., Wang, L. (2010). Dynamics of a stage-structured predator-prey model with prey impulsively diffusive between two patches. Nonlinear Analysis: Real World Applications, 11, 2748–2756.
Li, K., & Wei, J. (2009). Stability and Hopf bifurcation analysis of a prey-predator system with two delays. Chaos, Solitons and Fractals, 42(5), 2606–2613.
Song, Y., Peng, Y., Wei, J. (2008). Bifurcations for a predator-prey system with two delays. Journal of Mathematical Analysis and Applications, 337(1), 466–479.
Thomas, M.B., Wood, S.N., Lomer, C.J. (1995). Biological control of locusts and grasshoppers using a fungal pathogen: the importance of secondary cycling. Proceedings of the Royal Society of London, Series B: Biological Sciences, 259(1356), 265–270.
Xu, C., Liao, M., He, X. (2011). Stability and Hopf bifurcation analysis for a Lotka–Volterra predator-prey model with two delays. International Journal of Applied Mathematics and Computer Science, 21(1), 97–107.
Wu, T. (2013). Study of the a impulsive prey-predator harvesting model with Beddington-Deangelis functional response. Advanced Materials Research, 616, 2060–2063.
Lian, F., & Xu, Y. (2009). Hopf bifurcation analysis of a predator–prey system with Holling type-IV functional response and time delay. Applied Mathematics and Computation, 215(4), 1484–1495.
Liu, X., & Han, M. (2011). Chaos and Hopf bifurcation analysis for a two species predator-prey system with prey refuge and diffusion. Nonlinear Analysis: Real World Applications, 12(2), 1047–1061.
Zhao, H., & Lin, Y. (2009). Hopf bifurcation in a partial dependent predator-prey system with delay. Chaos, Solitons and Fractals, 42(2), 896–900.
Gazzoni, D.L. (1994). Manejo de pragas da soja: uma abordagem historica. EMBRAPACNPSO: Londrina.
DeBach, P., & Rosen, D. (1991). Biological control by natural enemies. Cambridge: Cambridge University Press.
Luff, M.L. (1983). The potential of predators for pest control. Agriculture, Ecosystems and Environment, 10 (2), 159–181.
Rutledge, C.E., O’Neil, R.J., Fox, T.B., Landis, D.A. (2004). Soybean aphid predators and their use in integrated pest management. Annals of the Entomological Society of America, 97(2), 240–248.
Dhar, J., & Jatav, K.S. (2013). Mathematical analysis of a delayed stage-structured predator-prey model with impulsive diffusion between two predators territories. Ecological Complexity, 16, 59–67.
Driver, R.D. (1977). Ordinary and delay differential equations Vol. 20. New York: Springer.
Singh, H., Dhar, J., Bhatti, H.S. (2016). Dynamics of a prey-generalized predator system with disease in prey and gestation delay for predator. Modeling Earth Systems and Environment, 2, 52. https://doi.org/10.1007/s40808-016-0096-8.
Ruan, S. (2001). Absolute stability, conditional stability and bifurcation in Kolmogorov-type predator-prey systems with discrete delays. Quarterly of Applied Mathematics, 59(1), 159–174.
Singh, H., Dhar, J., Bhatti, H.S. (2016). An epidemic model of childhood disease dynamics with maturation delay and latent period of infection. Modeling Earth Systems and Environment, 2, 79. https://doi.org/10.1007/s40808-016-0131-9.
Bera, S.P., Maiti, A., Samanta, G.P. (2016). Dynamics of a food chain model with herd behaviour of the prey. Modeling Earth Systems and Environment, 2, 131. https://doi.org/10.1007/s40808-016-0189-4.
Acknowledgements
I would like to thank the I.K.G.-Punjab Technical University, Kapurthala 144601, Punjab, India.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendices
Appendix: I
Let (Pi(t),Pm(t),N(t)) be the solution of the proposed (1)–(3) with nonnegative initial populations. For t ∈ [0,τ], the Eq. 2, we get the following:
which evidences that
Also, for t ∈ [0,τ], the Eq. 1 is as follows:
since the mature pest population is nonnegative and monotonic increasing function of t in [−τ, 0] and using (6), we get the following:
it follows that
Finally, for t ∈ [0,τ], from the Eq. 3, we have the following:
which results that
Similarly, for the intervals [τ, 2τ],..., [nτ, (n + 1)τ], n ∈N, it can be proved that Pi(t), Pm(t) and N(t) all are nonnegative. Hence, from Eqs. 6–8, the population remains positive, i.e., Pi(t) ≥ 0, Pm(t) ≥ 0 and N(t) ≥ 0 for all t ≥ 0.
Appendix II
Assume that V (t) = Pi + Pm + N. Now differentiating V (t) w.r.t. t, we have as follows:
Taking c1 = min{μ1,μ2,μ3}, we have as follows:
There exists a constant \(c_{2}=\frac {\alpha ^{2}}{4 \beta _{1}}\), which is positive, therefore
Thus, using Pi(t) ≥ 0, Pm(t) ≥ 0 and N(t) ≥ 0 for all t ≥ 0, we get the following:
As t →∞, we get the following:
Clearly, the system (1)–(3), is bounded above for its each species.
Appendix III
-
(i)
The characteristic equation for E0(0, 0, 0) is
$$ (\lambda+\mu_{1})(\lambda-\alpha e^{-\mu_{1}\tau}e^{-\lambda\tau}+\mu_{2})(\lambda+\mu_{3})= 0. $$(9)Here, λ = −μ1,λ = −μ3 are two negative roots of Eq. 9. If \(\alpha e^{-\mu _{1}\tau }<\mu _{2}\), then the third root of Eq. 9 is also negative. Therefore, the equilibrium point E0(0, 0, 0) is locally asymptotically stable, if \(\alpha e^{-\mu _{1}\tau }<\mu _{2}\), that is, if
$$\tau>\frac{1}{\mu_{1}}\ln\left( \frac{\alpha}{\mu_{2}}\right)=\tau_{0}.$$Thus, if τ > τ0, then the equilibrium E0(0, 0, 0) is locally asymptotically stable for all τ. Moreover, E0(0, 0, 0) is unstable for all τ < τ0, because one of the root of Eq. 9 has positive real part.
-
(ii)
The characteristic equation for E1(Pi1,Pm1, 0) is as follows:
$$ (-\lambda-\mu_{1})F_{1}(\lambda)= 0, $$(10)where \(F_{1}(\lambda )=\lambda ^{2}+\lambda (\mu _{2}+ 2\beta _{1} P_{m}-\frac {\gamma _{1} P_{m}}{(a+P_{m})}+\mu _{3})+(2 \beta _{1} P_{m} \mu _{3}+ \mu _{2} \mu _{3}-\mu _{2} \frac {\gamma _{1} P_{m}}{a+P_{m}}-\frac {2\beta _{1} \gamma _{1} {P_{m}^{2}} }{a+P_{m}}+(\alpha e^{-\mu _{1} \tau } (\frac {\gamma _{1} P_{m}}{a+P_{m}}-\mu _{3})-\alpha e^{-\mu _{1} \tau }\lambda )e^{-\lambda \tau }\),
Clearly, F1(λ) = 0 is of the form (5), i.e., λ2 + pλ + r + (sλ + q)e−λτ = 0, here, \(p=\mu _{2}+ 2\beta _{1} P_{m}-\frac {\gamma _{1} P_{m}}{(a+P_{m})}+\mu _{3}\), \(r = 2 \beta _{1} P_{m} \mu _{3}+ \mu _{2} \mu _{3}-\mu _{2} \frac {\gamma _{1} P_{m}}{a+P_{m}}-\frac {2\beta _{1} \gamma _{1} {P_{m}^{2}} }{a+P_{m}}\), \(s=-\alpha e^{-\mu _{1} \tau }\) and \(q=\alpha e^{-\mu _{1} \tau } (\frac {\gamma _{1} P_{m}}{a+P_{m}}-\mu _{3})\).
-
Case I: When τ = 0, we get the following:
$$ \lambda^{2}+(p+s)\lambda+(q+r)= 0. $$(11)If the conditions [B1] and [B2] hold, then the Eq. 11 has negative real roots. Therefore, the steady state E1(Pi1,Pm1, 0) is locally asymptotically stable.
-
Case II: If τ > 0, we have as follows:
$$ {\lambda }^{2}+p\lambda+r+(s\lambda +q)e^{-\lambda \tau}= 0, $$(12)
Again, if the conditions [B1], [B2], and [B3] hold, then by Lemma 3, the proposed system (1)–(3) has negative real roots for all τ; hence, the system becomes locally asymptotically stable. Furthermore, by Lemma 3, if the conditions [B1], [B2], and [B4] hold, then the proposed system (1)–(3) has a pair of purely imaginary roots. Substitute λ = iw in Eq. 12, we must have the following:
$$ {(iw) }^{2}+p(iw)+r+(iws +q)e^{-iw \tau}= 0. $$(13)Comparing real and imaginary parts from Eq. 13,
$$\begin{array}{@{}rcl@{}} -w^{2}+r+sw\sin {w\tau}+q\cos {w\tau}= 0, \end{array} $$(14)$$\begin{array}{@{}rcl@{}} pw+sw\cos {w\tau}-q\sin {w\tau}= 0. \end{array} $$(15)$$\begin{array}{@{}rcl@{}} \sin w\tau=\frac{sw^{3}+(pq-rs)w}{s^{2}w^{2}+q^{2}}, \end{array} $$(16)$$\begin{array}{@{}rcl@{}} \cos w\tau=\frac{(q-ps)w^{2}-qr}{s^{2}w^{2}+q^{2}} \end{array} $$(17)and
$$\begin{array}{@{}rcl@{}} w^{4}+(p^{2}-2r-s^{2})w^{2}+(r^{2}-q^{2})= 0. \end{array} $$(18)Now, let us take the following:
$$F(w)=w^{4}+(p^{2}-2r-s^{2})w^{2}+(r^{2}-q^{2})= 0.$$According to Descart’s rule of sign, there exists at least one nonnegative root of F(w) = 0. Let w0 > 0 be such root of F(w) = 0. From Eq. 17,
$$\tau_{k}^{+}=\frac{1}{w_{0}}\left[\cos^{-1}\left( \frac{(q-ps){w_{0}}^{2}-qr}{s^{2}{w_{0}}^{2}+q^{2}}\right)+ 2k\pi\right], \text{where} \ k = 0,1,2,.... $$Since, for Hopf bifurcation is to exist at \(\tau _{0}^{+}\), the condition \(Re\left [\left (\frac {d\lambda }{d\tau }\right )^{-1}\right ]_{\tau =\tau _{0}^{+}}\neq 0\) should be true, therefore differentiating λ w.r.t. τ in Eq. 12, we have as follows:
$$\frac{d\lambda}{d\tau}=\frac{\lambda(s\lambda+q)e^{-\lambda \tau}}{2\lambda+p+se^{-\lambda \tau}-(s\lambda+q)\tau e^{-\lambda \tau}}.$$Substitute λ = iw0 and \(\tau =\tau _{0}^{+}\), we get the following:
$$ Re\left( \frac{d\lambda}{d\tau}\right)^{-1}=\frac{qG-sw_{0}H}{w_{0}(q^{2}+s^{2}{w_{0}^{2}})}, $$(19)where G = psinw0τ0 + 2w0cosw0τ0 and H = s + pcosw0τ0 − 2w0sinw0τ0. Solving Eq. 19, we have as follows:
$$Re\left[\left( \frac{d\lambda}{d\tau}\right)^{-1}\right]_{\tau=\tau_{0}^{+}}\neq0, \ \ \text{if} \ qG\neq sw_{0}H. $$ -
-
(iii)
The characteristic equation at the interior equilibrium point E∗ is of the form:
$$ (-\lambda-\mu_{1})F_{2}(\lambda)= 0, $$(20)where
$$\begin{array}{@{}rcl@{}} F_{2}(\lambda) &=& \lambda^{2}+\lambda \left( \mu_{2}+ \mu_{3} + \frac{\gamma a N^{*}}{(a+P_{m}^{*})^{2}}+ 2\beta_{1} P_{m}^{*} \right)\\&&+ \mu_{3} \mu_{2} + \mu_{3} \frac{\gamma a N^{*}}{(a+P_{m}^{*})^{2}} \\ &&+ \mu_{3} 2\beta_{1} P_{m}^{*} + \frac{\gamma \gamma_{1} a N^{*} P_{m}^{*}}{(a+P_{m}^{*})^{3}} \\&&+ (-\lambda\alpha e^{-\mu_{1} \tau} - \mu_{3} \alpha e^{-\mu_{1} \tau})e^{-\lambda \tau}, \end{array} $$on comparing F2(λ) = 0 with Eq. 5, we have as follows: λ2 + pλ + r + (sλ + q)e−λτ = 0, here, \(p=\mu _{2} + \mu _{3} + \frac {\gamma a N^{*}}{(a+P_{m}^{*})^{2}}+ 2\beta _{1} P_{m}^{*}\), \(r=\mu _{3} \left (\mu _{2} + \frac {\gamma a N^{*}}{(a+P_{m}^{*})^{2}}+ 2\beta _{1} P_{m}^{*}\right ) + \frac {\gamma \gamma _{1} a N^{*} P_{m}^{*}}{(a+P_{m}^{*})^{3}}\), \(s=-\alpha e^{-\mu _{1} \tau }\) and \(q=- \mu _{3} \alpha e^{-\mu _{1} \tau }\).
-
Case I: If τ = 0, we obtain the following:
$$ \lambda^{2}+(p+s)\lambda+(q+r)= 0. $$(21)If the conditions [B1] and [B2] hold, then the Eq. 21 has negative real roots. Therefore, the steady state E∗ is locally asymptotically stable.
-
Case II: For τ > 0, a part of Eq. 20, i.e., F2(λ) = 0 can be written as follows:
$$ {\lambda }^{2}+p\lambda+r+(s\lambda +q)e^{-\lambda \tau}= 0, $$(22)
If the conditions [B1], [B2], and [B3] hold, then using Lemma 3, the proposed system (1)–(3) has nonpositive real roots for every τ; hence, the model becomes locally asymptotically stable. Again, by Lemma 3, if [B1], [B2], and [B4] hold, then the model (1)–(3) has a pair of purely imaginary roots. Taking λ = iw in Eq. 22, we obtain the following:
$$ {(iw) }^{2}+p(iw)+r+(iws +q)e^{-iw \tau}= 0. $$(23)Comparing real and imaginary parts from Eq. 23, we obtain the following;
$$\begin{array}{@{}rcl@{}} -w^{2}+r+sw\sin {w\tau}+q\cos {w\tau}= 0, \end{array} $$(24)$$\begin{array}{@{}rcl@{}} pw+sw\cos {w\tau}-q\sin {w\tau}= 0. \end{array} $$(25)Simplifying (24) and (25), we have as follows:
$$\begin{array}{@{}rcl@{}} \sin w\tau=\frac{sw^{3}+(pq-rs)w}{s^{2}w^{2}+q^{2}}, \end{array} $$(26)$$\begin{array}{@{}rcl@{}} \cos w\tau=\frac{(q-ps)w^{2}-qr}{s^{2}w^{2}+q^{2}} \end{array} $$(27)and
$$\begin{array}{@{}rcl@{}} w^{4}+(p^{2}-2r-s^{2})w^{2}+(r^{2}-q^{2})= 0. \end{array} $$(28)Now, let us take the following:
$$F(w)=w^{4}+(p^{2}-2r-s^{2})w^{2}+(r^{2}-q^{2})= 0.$$According to Descart’s rule of signs, there exists at least one positive root of F(w) = 0. Let it be w0 > 0, for F(w) = 0. From Eq. 27, we have as follows:
$$\tau_{k}^{+}=\frac{1}{w_{0}}\left[\cos^{-1}\left( \frac{(q-ps){w_{0}}^{2}-qr}{s^{2}{w_{0}}^{2}+q^{2}}\right)+ 2k\pi\right], \ \text{where} \ k = 0,1,2,.... $$For Hopf bifurcation is to take place at \(\tau _{0}^{+}\), the condition \(Re\left [\left (\frac {d\lambda }{d\tau }\right )^{-1}\right ]_{\tau =\tau _{0}^{+}}\neq 0\) should hold, therefore differentiating λ w.r.t. τ in Eq. 22 as follows:
$$\frac{d\lambda}{d\tau}=\frac{\lambda(s\lambda+q)e^{-\lambda \tau}}{2\lambda+p+se^{-\lambda \tau}-(s\lambda+q)\tau e^{-\lambda \tau}}.$$Put λ = iw0 and \(\tau =\tau _{0}^{+}\), we obtain the following:
$$ Re\left( \frac{d\lambda}{d\tau}\right)^{-1}=\frac{q G_{1}-sw_{0} H_{1}}{w_{0}(q^{2}+s^{2}{w_{0}^{2}})}, $$(29)where G1 = psinw0τ0 + 2w0cosw0τ0 and H1 = s + pcosw0τ0 − 2w0sinw0τ0. Solving the Eq. 29, we have as follows:
$$Re\left[\left( \frac{d\lambda}{d\tau}\right)^{-1}\right]_{\tau=\tau_{0}^{+}}\neq0, \ \ \text{if} \ q G_{1}\neq sw_{0} H_{1}. $$ -
Rights and permissions
About this article
Cite this article
Kumar, V., Dhar, J. & Bhatti, H.S. A Stage-Structured Pest-Natural Enemy Dynamics with Holling Type-II Interaction and Maturation Delay for Pest Species. Environ Model Assess 24, 355–363 (2019). https://doi.org/10.1007/s10666-019-9652-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10666-019-9652-8