# Effect of Slow–Fast Time Scale on Transient Dynamics in a Realistic Prey-Predator System

^{1}

^{2}

^{3}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. The Slow–Fast Model

## 3. Bifurcation Results

## 4. Slow–Fast Dynamics

**Lemma**

**1.**

**Proof.**

**Theorem**

**1.**

**Proof.**

## 5. Transient Dynamics

## 6. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Rinaldi, S.; Muratori, S. Slow-fast limit cycles in predator-prey models. Ecol. Model.
**1992**, 61, 287–308. [Google Scholar] [CrossRef] - Rinaldi, S.; Scheffer, M. Geometric analysis of ecological models with slow and fast processes. Ecosystems
**2000**, 3, 507–521. [Google Scholar] [CrossRef] - Fenichel, N. Geometric Singular Perturbation Theory for Ordinary Differential Equations. J. Differ. Equ.
**1979**, 31, 53–98. [Google Scholar] [CrossRef] [Green Version] - Krupa, M.; Szmolyan, P. Extending Geometric Singular Perturbation Thoery to nonhyperbolic points- folds and canards in two dimension. SIAM J. Math. Anal.
**2001**, 33, 286–314. [Google Scholar] [CrossRef] [Green Version] - Krupa, M.; Szmolyan, P. Relaxation oscillation and Canard explosion. J. Differ. Equ.
**2001**, 174, 312–368. [Google Scholar] [CrossRef] [Green Version] - Kooi, B.W.; Poggiale, J.C. Modelling, singular perturbation and bifurcation analyses of bitrophic food chains. Math. Biosci.
**2018**, 301, 93–110. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Chowdhury, P.R.; Petrovskii, S.; Banerjee, M. Oscillations and Pattern Formation in a Slow–Fast Prey–Predator System. Bull. Math. Biol.
**2021**, 83, 110. [Google Scholar] [CrossRef] [PubMed] - Hastings, A.; Abbott, K.C.; Cuddington, K.; Francis, T.; Gellner, G.; Lai, Y.C.; Morozov, A.; Petrovskii, S.; Scranton, K.; Zeeman, M.L. Transient phenomena in ecology. Science
**2018**, 7, eaat6412. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Hastings, A. Transients: The key to long-term ecological understanding? Trends Ecol. Evol
**2004**, 19, 39–45. [Google Scholar] [CrossRef] [PubMed] - Hastings, A. Transient dynamics and persistence of ecological systems. Ecol. Lett.
**2001**, 4, 215–220. [Google Scholar] [CrossRef] - Morozov, A.; Banerjee, M.; Petrovskii, S. Long-term transients and complex dynamics of a stage-structured population with time delay and the Allee effect. J. Theor. Biol.
**2016**, 396, 116–124. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Morozov, A.; Abbott, K.C.; Cuddington, K.; Francis, T.; Gellner, G.; Hastings, A.; Lai, Y.C.; Petrovskii, S.V.; Scranton, K.; Zeeman, M.L. Long transients in ecology: Theory and applications. Phys. Life Rev.
**2020**, 32, 1–40. [Google Scholar] [CrossRef] [PubMed] - Arditi, R.; Ginzburg, L.R. Coupling in predator-prey dynamics: Ratio-dependence. J. Theor. Biol.
**1989**, 139, 311–326. [Google Scholar] [CrossRef] - Bazykin, A.; Khibnik, A.; Krauskopf, B. Nonlinear Dynamics of Interacting Populations; World Scientific: Singapore, 1998; Volume 11. [Google Scholar]
- Banerjee, M.; Petrovskii, S. Self-Organised spatial patterns and chaos in a ratio-dependent predator-prey system. Theor. Ecol.
**2011**, 4, 37–53. [Google Scholar] [CrossRef] - Banerjee, M.; Abbas, S. Existence and non-existence of spatial patterns in a ratio-dependent predator–prey model. Eco. Complex.
**2015**, 21, 199–214. [Google Scholar] [CrossRef] - Arancibia-Ibarra, C.; Aguirre, P.; Flores, J.; Heijste, P. Bifurcation analysis of a predator-prey model with predator intraspecific interactions and ratio-dependent functional response. Appl. Math. Comput.
**2021**, 402, 126152. [Google Scholar] [CrossRef] - Kuznetsov, Y.A. Elements of Applied Bifurcation Theory; Springer: New York, NY, USA, 2004. [Google Scholar]
- Seydel, R. Practical Bifurcation and Stability Analysis; Springer: Berlin/Heidelberg, Germany, 2009. [Google Scholar]
- Kuehn, C. Multiple Time Scale Dynamics; Springer: New York, NY, USA, 2015. [Google Scholar]
- Sadhu, S. Analysis of long-term transients and detection of early warning signs of major population changes in a two-timescale ecosystem. arXiv
**2021**, arXiv:2105.09411. [Google Scholar] - Saha, T.; Pal, P.J.; Banerjee, M. Relaxation oscillation and22 canard explosion in a slow-fast predator-prey model with Beddington-DeAngelis functional response. Nonlinear Dyn.
**2021**, 103, 1195–1217. [Google Scholar] [CrossRef]

**Figure 1.**Two parametric global bifurcation diagram of system (1) for $\alpha =2,\phantom{\rule{4pt}{0ex}}\gamma =0.6,$ and $\epsilon =1$. ${\beta}_{GH},\phantom{\rule{4pt}{0ex}}{\beta}_{TC},and\phantom{\rule{4pt}{0ex}}{\beta}_{HT}$ represent the $\beta $-threshold for generalized Hopf bifurcation, transcritical bifurcation, and Bogdanov–Takens bifurcation, respectively. The point CP is the cusp point, the intersection of the vertical line ${\beta}_{TC}$ with the saddle node bifurcation curve of equilibrium points.

**Figure 3.**(

**a**) Canard cycle without head (green) for $\delta =0.111,$ canard cycle with head for $\delta =0.1104,$ relaxation oscillation for $\delta =0.09.$ (

**b**) The bifurcation diagram the change in the amplitude of the limit cycle. Canard explosion occurs in the pink shaded region. The other parameter values are fixed at $\alpha =2,\phantom{\rule{4pt}{0ex}}\gamma =0.6,\phantom{\rule{4pt}{0ex}}\beta =0.88$, and $\epsilon =0.05.$.

**Figure 4.**(

**a**) Stable canard cycle with head (blue) and unstable canard cycle (red) for $\beta =1.15,\phantom{\rule{4pt}{0ex}}\delta =0.158$, and $\epsilon =0.1.$ Green dot represents stable equilibrium point. (

**b**) The bifurcation diagram shows the change in the amplitude of v-component of the limit cycle. The solid line represents the maximum and minimum amplitude of limit cycles, stable (blue), and unstable (red). The broken line shows the unique interior component where red represents unstable and blue is for stable.

**Figure 5.**Relaxation oscillation ${\gamma}_{\epsilon}$ for $\epsilon =0.05$ (cyan) and the singular slow–fast trajectory ${\gamma}_{0}$ (broken blue curve), limit cycle (magenta) for $\epsilon =1$. Other parameters are fixed at $\beta =1.21,\phantom{\rule{4pt}{0ex}}\delta =0.8.$.

**Figure 6.**Solution of the system (1) showing long transient dynamics when approaching the attractor, as obtained numerically for three different values of $\epsilon $: $\epsilon =1$ (

**left**), $\epsilon =0.5$ (

**middle**), $\epsilon =0.1$ (

**right**). Other parameter values are $\alpha =2,\phantom{\rule{4pt}{0ex}}\gamma =0.6,\phantom{\rule{4pt}{0ex}}\beta =0.85,\phantom{\rule{4pt}{0ex}}\delta =0.08$.

**Figure 7.**The phase space (

**upper panel**) and the corresponding dependence of the prey density on time (

**lower panel**) showing the transient dynamics for $\beta =1.008502,\phantom{\rule{4pt}{0ex}}\delta =0.35.$ (

**a**,

**c**) $\epsilon =0.53,\phantom{\rule{4pt}{0ex}}$ (

**b**,

**d**) $\epsilon =0.05.$ Red dot represent the unstable equilibrium point. Other parameters are fixed as above.

**Figure 8.**(

**a**) The prey density as a function of time obtained for $\beta =1.24,\phantom{\rule{4pt}{0ex}}\delta =1.2$, and three values of $\epsilon $ (which are marked in the figure legend); (

**b**,

**c**) phase space and corresponding plot of the prey density for $\beta =1.239254,\phantom{\rule{4pt}{0ex}}\epsilon =0.52.$ The red and green dot represent the saddle and stable equilibrium point, respectively. Other parameters are fixed as in Figure 1.

**Figure 9.**Solution of system (1) showing the prey density as obtained for $\beta =1.21,\phantom{\rule{4pt}{0ex}}\delta =0.847,\phantom{\rule{4pt}{0ex}}\epsilon =0.9135$ with different initial data (

**a**) $(0.2,0.2)$ and (

**b**) $(0.14,0.104).$ (

**c**) Zoomed version of the initial transients before converging to periodic orbit for $\epsilon =0.9135,$ (

**left**) $\epsilon =0.9$ (

**middle**), and $\epsilon =0.1$ (

**right**), respectively.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Chowdhury, P.R.; Petrovskii, S.; Banerjee, M.
Effect of Slow–Fast Time Scale on Transient Dynamics in a Realistic Prey-Predator System. *Mathematics* **2022**, *10*, 699.
https://doi.org/10.3390/math10050699

**AMA Style**

Chowdhury PR, Petrovskii S, Banerjee M.
Effect of Slow–Fast Time Scale on Transient Dynamics in a Realistic Prey-Predator System. *Mathematics*. 2022; 10(5):699.
https://doi.org/10.3390/math10050699

**Chicago/Turabian Style**

Chowdhury, Pranali Roy, Sergei Petrovskii, and Malay Banerjee.
2022. "Effect of Slow–Fast Time Scale on Transient Dynamics in a Realistic Prey-Predator System" *Mathematics* 10, no. 5: 699.
https://doi.org/10.3390/math10050699