Skip to main content
Log in

A Stage-Structured Pest-Natural Enemy Dynamics with Holling Type-II Interaction and Maturation Delay for Pest Species

  • Published:
Environmental Modeling & Assessment Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

References

  1. Thomas, M.B., & Willis, A.J. (1998). Biocontrol-risky but necessarys. Trends in Ecology and Evolution, 13, 325–329.

    Article  CAS  Google Scholar 

  2. 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.

    Article  Google Scholar 

  3. 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.

    Article  CAS  Google Scholar 

  4. 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.

    Article  Google Scholar 

  5. Ang, L, Dongfang, X., Song, Y. (2016). https://doi.org/10.12783/dtetr/iect2016/3715.

  6. Aiello, W.G., & Freedman, H.I. (1990). A time-delay model of single-species growth with stage-structure. Mathematical Biosciences, 101(2), 139–153.

    Article  CAS  Google Scholar 

  7. 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.

    Article  Google Scholar 

  8. 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.

    Article  Google Scholar 

  9. 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.

    Article  Google Scholar 

  10. 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.

  11. 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.

    Article  Google Scholar 

  12. 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.

    Article  Google Scholar 

  13. 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.

    Article  Google Scholar 

  14. 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.

    Article  Google Scholar 

  15. 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.

    Article  Google Scholar 

  16. 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.

    Article  Google Scholar 

  17. 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.

    Article  Google Scholar 

  18. Wu, T. (2013). Study of the a impulsive prey-predator harvesting model with Beddington-Deangelis functional response. Advanced Materials Research, 616, 2060–2063.

    Google Scholar 

  19. 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.

    Article  Google Scholar 

  20. 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.

    Article  Google Scholar 

  21. Zhao, H., & Lin, Y. (2009). Hopf bifurcation in a partial dependent predator-prey system with delay. Chaos, Solitons and Fractals, 42(2), 896–900.

    Article  Google Scholar 

  22. Gazzoni, D.L. (1994). Manejo de pragas da soja: uma abordagem historica. EMBRAPACNPSO: Londrina.

    Google Scholar 

  23. DeBach, P., & Rosen, D. (1991). Biological control by natural enemies. Cambridge: Cambridge University Press.

    Google Scholar 

  24. Luff, M.L. (1983). The potential of predators for pest control. Agriculture, Ecosystems and Environment, 10 (2), 159–181.

    Article  Google Scholar 

  25. 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.

    Article  Google Scholar 

  26. 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.

    Article  Google Scholar 

  27. Driver, R.D. (1977). Ordinary and delay differential equations Vol. 20. New York: Springer.

    Book  Google Scholar 

  28. 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.

    Article  Google Scholar 

  29. 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.

    Article  Google Scholar 

  30. 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.

    Article  Google Scholar 

  31. 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.

    Article  Google Scholar 

Download references

Acknowledgements

I would like to thank the I.K.G.-Punjab Technical University, Kapurthala 144601, Punjab, India.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Vijay Kumar.

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:

$$\frac{dP_{m}}{dt}\geq -\mu_{2} P_{m}- \frac{\gamma P_{m} N}{a +P_{m}} -\beta_{1} {P_{m}^{2}},$$

which evidences that

$$ P_{m}(t)\geq \frac{P_{m}(0) e^{-\mu_{2} t}}{\exp \left\{ {{\int}_{0}^{t}} \left[\beta_{1} P_{m}(u) + \frac{\gamma N(u)}{a+P_{m}(u)} \right] du \right\}} = P_{m}^{*}(t)> 0. $$
(6)

Also, for t ∈ [0,τ], the Eq. 1 is as follows:

$$\frac{dP_{i}}{dt} = \alpha P_{m} - \mu_{1} P_{i}-\alpha e^{-\mu_{1} \tau} P_{m}(t-\tau),$$

since the mature pest population is nonnegative and monotonic increasing function of t in [−τ, 0] and using (6), we get the following:

$$\frac{dP_{i}}{dt}\geq - \mu_{1} P_{i} +\alpha (1- e^{-\mu_{1} \tau}) P_{m}^{*}(t), $$

it follows that

$$ P_{i}(t)\geq \frac{P_{i}(0) + {{\int}_{0}^{t}}{\alpha(1-e^{-\mu_{1} \tau})e^{\mu_{1} s} P_{m}^{*}(s) ds}}{e^{\mu_{1} t}} > 0. $$
(7)

Finally, for t ∈ [0,τ], from the Eq. 3, we have the following:

$$\frac{dN}{dt}=\frac{\gamma_{1} P_{m} N}{a+P_{m}}-\mu_{3} N,$$

which results that

$$ N(t)= N(0) \ {\exp \left\{ {{\int}_{0}^{t}} \left[\frac{\gamma_{1} P_{m}(v)}{a+P_{m}(v)}-\mu_{3} \right] dv \right\}}> 0. $$
(8)

Similarly, for the intervals [τ, 2τ],..., [nτ, (n + 1)τ], nN, it can be proved that Pi(t), Pm(t) and N(t) all are nonnegative. Hence, from Eqs. 68, 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:

$$\begin{array}{@{}rcl@{}} \frac{dV(t)}{dt} &=& \alpha P_{m}-\mu_{1} P_{i}-\mu_{2} P_{m}-\frac{\gamma P_{m} N}{a+P_{m}}-\beta_{1} {P_{m}^{2}}+\frac{\gamma_{1} P_{m} N}{a+P_{m}}-\mu_{3} N,\\ &\leq & \alpha P_{m}-\mu_{1} P_{i}-\mu_{2} P_{m}-\beta_{1} {P_{m}^{2}}-\mu_{3}N, \ \text{where} \ \gamma_{1}<<\gamma. \end{array} $$

Taking c1 = min{μ1,μ2,μ3}, we have as follows:

$$\frac{dV(t)}{dt}+c_{1}V\leq\alpha P_{m}-\beta_{1} {P_{m}^{2}}.$$

There exists a constant \(c_{2}=\frac {\alpha ^{2}}{4 \beta _{1}}\), which is positive, therefore

$$\frac{dV(t)}{dt}+c_{1}V\leq c_{2}.$$

Thus, using Pi(t) ≥ 0, Pm(t) ≥ 0 and N(t) ≥ 0 for all t ≥ 0, we get the following:

$$0 \leq V(t)\leq V(0)e^{-c_{1}t}+\frac{c_{2}}{c_{1}}.$$

As t, we get the following:

$$0\leq V(t)\leq \frac{c_{2}}{c_{1}}.$$

Clearly, the system (1)–(3), is bounded above for its each species.

Appendix III

  1. (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.

  2. (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)eλτ = 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)

    Simplifying (14) and (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:

    $$Re\left[\left( \frac{d\lambda}{d\tau}\right)^{-1}\right]_{\tau=\tau_{0}^{+}}\neq0, \ \ \text{if} \ qG\neq sw_{0}H. $$
  3. (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)eλτ = 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

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

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

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10666-019-9652-8

Keywords

Navigation