# Stability Switching in Lotka-Volterra and Ricker-Type Predator-Prey Systems with Arbitrary Step Size

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## Abstract

**:**

## 1. Introduction

## 2. Equilibrium Stability and Dynamics

- (i)
- $\left|{\lambda}_{i}\right|<1,\phantom{\rule{0.222222em}{0ex}}i=1,2$; a sink, locally asymptotically stable.
- (ii)
- $\left|{\lambda}_{i}\right|>1,\phantom{\rule{0.222222em}{0ex}}i=1,2$; a source.
- (iii)
- One of $\left|{\lambda}_{i}\right|>1$ and other $\left|{\lambda}_{i}\right|<1,\phantom{\rule{0.222222em}{0ex}}i=1,2$; a saddle.
- (iv)
- One of $\left|{\lambda}_{i}\right|=1$ and other $\left|{\lambda}_{i}\right|\ne 1,\phantom{\rule{0.222222em}{0ex}}i=1,2$; non-hyperbolic.

## 3. Deriving Connections of Dynamical Properties in Discrete and Continuous Systems

- (i)
- $\theta <-r\mathrm{and}\frac{2}{r}<h<\frac{2}{1-{e}^{\theta}},$
- (ii)
- $\theta >-r\mathrm{and}\frac{2}{r}>h>\frac{2}{1-{e}^{\theta}}$.

- (i)
- $\theta <ln(1-r)\mathrm{and}\frac{2}{r}<h<\frac{-2}{\theta},$
- (ii)
- $\theta >ln(1-r)\mathrm{and}\frac{2}{r}>h>\frac{-2}{\theta}.$

- (i)
- If ${\lambda}_{1}$ and ${\lambda}_{2}$ are real and negative where ${\lambda}_{1}={\lambda}_{2}$, then ${E}_{3}$ is asymptotically stable if$$h<\frac{T}{4}.$$This happens only if $T=4\theta $.
- (ii)
- If both ${\lambda}_{1}$ and ${\lambda}_{2}$ are real and negative where ${\lambda}_{1}\ne {\lambda}_{2}$, then ${E}_{3}$ is asymptotically stable if the step size satisfies$$h<\left\{-\frac{2}{{\lambda}_{1}},-\frac{2}{{\lambda}_{2}}\right\}.$$This happens if $T-4\theta >0$.
- (iii)
- If both ${\lambda}_{1}$ and ${\lambda}_{2}$ are complex conjugate eigenvalues (say $a\pm ib$), then, from the above, ${a}^{2}+{b}^{2}=D=\theta T$ and $T=-2a$. This occurs when $T-4\theta <0$. With these complex eigenvalues, the population dynamics lead to oscillations with time. Then, the bound for the step size is$$h<\left\{\frac{-2a}{{a}^{2}+{b}^{2}}\right\}=\frac{T}{D}=\frac{1}{\theta}.$$This can only happen if $0<1+hT(h\theta -1)$. Note that, if $h=\frac{1}{\theta}$, then both eigenvalues have magnitude one.

## 4. Numerical Results

## 5. Discussion and Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

ODE | Ordinary differential equations |

RK | Ricker-type |

LV | Lotka-Volterra |

## References

- Ricker, W.E. Handbook of computations for biological statistics of fish populations. Bull. Fish. Res. Board Can.
**1958**, 119, 300. [Google Scholar] - Baxter, P.W.J.; Sabo, J.L.; Wilcox, C.; McCarthy, M.A.; Possingham, H.P. Cost-Effective Suppression and Eradication of Invasive Predators. Conserv. Biol.
**2008**, 22, 89–98. [Google Scholar] [CrossRef] [PubMed] - Sabo, J.L. Stochasticity, predator–prey dynamics, and trigger harvest of nonnative predators. Ecology
**2005**, 86, 2329–2343. [Google Scholar] [CrossRef] - Enatsu, Y.; Nakata, Y.; Muroya, Y.; Izzo, G.; Vecchio, A. Global dynamics of difference equations for SIR epidemic models with a class of nonlinear incidence rates. J. Differ. Equ. Appl.
**2012**, 18, 1163–1181. [Google Scholar] [CrossRef] - Jana, D. Chaotic dynamics of a discrete predator–prey system with prey refuge. Appl. Math. Comput.
**2013**, 224, 848–865. [Google Scholar] [CrossRef] - Wang, X.; Cheng, J.; Wang, L. A reinforcement learning-based predator-prey model. Ecol. Complex.
**2020**, 42, 100815. [Google Scholar] [CrossRef] - Hines, K.E.; Middendorf, T.R.; Aldrich, R.W. Determination of parameter identifiability in nonlinear biophysical models: A Bayesian approach. J. Gen. Physiol.
**2014**, 143, 401–416. [Google Scholar] [CrossRef] - Kekulthotuwage Don, S.P. Novel Mathematical Models and Simulation Tools for Stochastic Ecosystems. Ph.D. Thesis, Queensland University of Technology, Queensland, Australia, 2022. [Google Scholar] [CrossRef]
- Liu, X.; Xiao, D. Complex dynamic behaviors of a discrete-time predator–prey system. Chaos Solitons Fractals
**2007**, 32, 80–94. [Google Scholar] [CrossRef] - Din, Q. Stability, bifurcation analysis and chaos control for a predator-prey system. J. Vib. Control
**2019**, 25, 612–626. [Google Scholar] [CrossRef] - Krivine, H.; Lesné, A.; Treiner, J. Discrete-time and continuous-time modelling: Some bridges and gaps. Math. Struct. Comput. Sci.
**2007**, 17, 261–276. [Google Scholar] [CrossRef] - Brauer, F.; Castillo-Chavez, C. Mathematical Models in Population Biology and Epidemiology; Springer: New York, NY, USA, 2012; Volume 2. [Google Scholar] [CrossRef]
- Zhao, J. Complexity and chaos control in a discrete-time Lotka–Volterra predator–prey system. J. Differ. Equ. Appl.
**2020**, 26, 1303–1320. [Google Scholar] [CrossRef] - Windarto, W.; Eridani, E. On modification and application of Lotka–Volterra competition model. AIP Conf. Proc.
**2020**, 2268, 050007. [Google Scholar] [CrossRef] - Ackleh, A.S.; Salceanu, P.L. Competitive exclusion and coexistence in an n-species Ricker model. J. Biol. Dyn.
**2015**, 9, 321–331. [Google Scholar] [CrossRef] [PubMed] - Merdan, H. Stability analysis of a Lotka–Volterra type predator–prey system involving Allee effects. ANZIAM J.
**2010**, 52, 139–145. [Google Scholar] [CrossRef] - Alligood, K.T.; Sauer, T.D.; Yorke, J.A. Two-Dimensional Maps. In Chaos: An Introduction to Dynamical Systems; Springer: New Delhi, India, 1996; pp. 43–104. [Google Scholar] [CrossRef]
- Din, Q. Dynamics of a discrete Lotka-Volterra model. Adv. Differ. Equ.
**2013**, 2013, 95. [Google Scholar] [CrossRef] - Merdan, H.; Duman, O. On the stability analysis of a general discrete-time population model involving predation and Allee effects. Chaos Solitons Fractals
**2009**, 40, 1169–1175. [Google Scholar] [CrossRef] - Efimov, D.; Polyakov, A.; Aleksandrov, A. Discretization of homogeneous systems using Euler method with a state-dependent step. Automatica
**2019**, 109, 108546. [Google Scholar] [CrossRef] - Seno, H. A discrete prey–predator model preserving the dynamics of a structurally unstable Lotka–Volterra model. J. Differ. Equ. Appl.
**2007**, 13, 1155–1170. [Google Scholar] [CrossRef] - Mickens, R.E. Dynamic consistency: A fundamental principle for constructing nonstandard finite difference schemes for differential equations. J. Differ. Equ. Appl.
**2005**, 11, 645–653. [Google Scholar] [CrossRef] - Rana, S.S. Chaotic dynamics and control in a discrete-time predator-prey system with Ivlev functional response. Netw. Biol.
**2020**, 10, 45–61. [Google Scholar] - Yousef, A. Stability and further analytical bifurcation behaviors of Moran–Ricker model with delayed density dependent birth rate regulation. J. Comput. Appl. Math.
**2019**, 355, 143–161. [Google Scholar] [CrossRef] - Luis, R.; Elaydi, S.; Oliveira, H. Stability of a Ricker-type competition model and the competitive exclusion principle. J. Biol. Dyn.
**2011**, 5, 636–660. [Google Scholar] [CrossRef] - Chaudhary, H.; Khan, A.; Nigar, U.; Kaushik, S.; Sajid, M. An Effective Synchronization Approach to Stability Analysis for Chaotic Generalized Lotka–Volterra Biological Models Using Active and Parameter Identification Methods. Entropy
**2022**, 24, 529. [Google Scholar] [CrossRef] [PubMed] - Tunç, O.; Tunç, C.; Yao, J.C.; Wen, C.F. New fundamental results on the continuous and discrete integro-differential equations. Mathematics
**2022**, 10, 1377. [Google Scholar] [CrossRef] - Tunç, O.; Atan, Ö.; Tunç, C.; Yao, J.C. Qualitative analyses of integro-fractional differential equations with Caputo derivatives and retardations via the Lyapunov–Razumikhin method. Axioms
**2021**, 10, 58. [Google Scholar] [CrossRef] - Luís, R.; Rodrigues, E. Local Stability in 3D Discrete Dynamical Systems: Application to a Ricker Competition Model. Discret. Dyn. Nat. Soc.
**2017**, 2017. [Google Scholar] [CrossRef]

**Figure 1.**Demonstration of the different dynamics that can arise with the discrete Ricker-type system (1) when the parameter $\alpha $ is varied. This system is solved with $K=2500,\gamma =0.01,c=0.2$ and $r=0.5$. For three different $\alpha $ values $0.05,0.048$ and $0.04$, the predator-prey populations diverge, converge very slowly and converge, respectively. Here, system (1) is derived for a unit step size, which is similar to system (2) when $h=1$.

**Figure 2.**Different predator-prey dynamics of discrete Ricker-type and Lotka-Volterra models with slightly varying parameter c as $c+\zeta $ by $\zeta =\{0,0.01,0.05,0.1,0.2\}$ values where $h=1,K=2500,\gamma =0.01,\alpha =0.05,c=0.2$ and $r=0.5$.

**Figure 3.**Stability regions of the discrete Ricker-type model and discrete Lotka-Volterra model as a function of h and $\beta =\alpha \gamma $, $K=2500$ and $r=0.5$. The fixed-point convergence region is bounded by $\beta =\frac{c}{K}+\frac{1}{h}$ and $\beta =\frac{c}{K}$, and boundary changes are marked in red and black lines for different $c=\{0.1,0.2,0.3\}$. The stability regions are coloured for $c=0.2$, as represented in solid lines, and are represented as dashed lines for $c=0.1$ and $c=0.3$.

**Figure 4.**The fixed-point convergence region as a function of $\beta $ and c for the discrete Ricker-type model and discrete Lotka-Volterra model where $\beta =\alpha \gamma $, $K=2500$ and $r=0.5$. The fixed-point convergence region is bounded by $\beta =\frac{c}{K}+\frac{1}{h}$ and $\beta =\frac{c}{K},\forall h>0$. For $h=1$, the stability regions are coloured, and the upper and lower boundaries of the fixed-point region are plotted for $\beta =\frac{c}{K}+1$ and $\beta =\frac{c}{K}$, as displayed in solid red and black lines, respectively. The upper boundary of fixed-point region moves upward with decreasing step size, marked as red dashed lines. Note that the lower boundary of the fixed-point convergence region is valid for any h.

**Figure 5.**Special behaviour of predator-prey populations for Ricker-type and Lotka-Volterra discrete models if $h=\frac{1}{\theta}=\frac{20}{21}$, where $K=2500,\gamma =0.01,\alpha =0.05,c=0.2$ and $r=0.5$. Predator-prey populations seem to converge to a fixed point at the beginning; however, after a long time, the populations oscillate around the fixed point. Note that this exceptional case occurs only at the upper bound of the fixed-point convergence region.

**Table 1.**Stability status for discrete and continuous Ricker-type (RK) and Lotka-Volterra (LV) models, where ${E}_{1}\equiv (0,0)$, ${E}_{2}\equiv (K,0),{E}_{3}\equiv \left(\frac{c}{\alpha \gamma},\frac{r}{\alpha}\left(1-\frac{c}{K\alpha \gamma}\right)\right)$, and ${\lambda}_{1},{\lambda}_{2}$ are eigenvalues of ${E}_{3}$ calculated from (13).

Stability | Model | ${\mathit{E}}_{\mathbf{1}}$ | ${\mathit{E}}_{\mathbf{2}}$ | ||

Discrete | Continuous | Discrete | Continuous | ||

Asym.stable | RK | - | - | if $\theta <0$ and $h<\{\frac{2}{r},\frac{2}{1-{e}^{\theta}}\}$ | if $\theta <0$ |

LV | - | - | if $\theta <0$ and $h<\{\frac{2}{r},-\frac{2}{\theta}\}$ | as above | |

Non-hyperbolic | RK | if $h=\frac{2}{1-{e}^{-c}}$ | - | if $\theta =0,h\ne \frac{2}{r}$ if $\theta <0,h=\frac{2}{1-{e}^{\theta}},h\ne \frac{2}{r}$ if $h=\frac{2}{r},\theta \ne 0,\theta \ne ln(1-r)$ | if $\theta =0$ |

LV | if $h=\frac{2}{c}$ | - | if $\theta =0,h\ne \frac{2}{r}$ if $\theta <0,h=-\frac{2}{\theta},\theta \ne 0,\theta \ne -r$ if $h=\frac{2}{r},\theta \ne 0,\theta \ne -r$ | as above | |

Saddle | RK | if $h<\frac{2}{1-{e}^{-c}}$ | always a saddle point | if $\theta <0,\theta <-r,\frac{2}{r}<h<\frac{2}{1-{e}^{\theta}}$ if $\theta <0,\theta >-r,\frac{2}{r}>h>\frac{2}{1-{e}^{\theta}}$ if $\theta >0,h<\{\frac{2}{r},\frac{2}{1-{e}^{\theta}}\}$ | if $\theta >0$ |

LV | if $h<\frac{2}{c}$ | always a saddle point | if $\theta <0,\theta <ln(1-r),\frac{2}{r}<h<-\frac{2}{\theta}$ if $\theta <0,\theta >ln(1-r),\frac{2}{r}>h>-\frac{2}{\theta}$ | as above | |

Stability | Model | ${\mathit{E}}_{\mathbf{3}}$ | |||

Discrete | Continuous | ||||

Asym.stable | RK | if $\theta >0$, $T=4\theta ,0<h<\frac{T}{4}$ if $\theta >0,T-4\theta >0,h<\{-\frac{2}{{\lambda}_{1}},-\frac{2}{{\lambda}_{2}}\}$ if $\theta >0,T-4\theta <0,h<\frac{1}{\theta},0<1+hT(h\theta -1)$ | if $\theta >0$ | ||

LV | as above | as above | |||

Non-hyperbolic | RK | if $\theta >0,T-4\theta >0,h=-\frac{2}{{\lambda}_{i}},h\ne -\frac{2}{{\lambda}_{j}},i\ne j,i,j=\{1,2\}$ if $\theta <0,h=-\frac{2}{{\lambda}_{2}},{\lambda}_{2}<0,{\lambda}_{1}>0$ if $\theta =0,h\ne \frac{2}{T}$ | if $\theta =0$ | ||

LV | as above | as above | |||

Saddle | RK | if $\theta >0,T-4\theta >0,-\frac{2}{{\lambda}_{i}}<h<-\frac{2}{{\lambda}_{j}},i\ne j,i,j=\{1,2\}$ | if $\theta <0$ | ||

LV | as above | as above |

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**MDPI and ACS Style**

Kekulthotuwage Don, S.; Burrage, K.; Helmstedt, K.J.; Burrage, P.M.
Stability Switching in Lotka-Volterra and Ricker-Type Predator-Prey Systems with Arbitrary Step Size. *Axioms* **2023**, *12*, 390.
https://doi.org/10.3390/axioms12040390

**AMA Style**

Kekulthotuwage Don S, Burrage K, Helmstedt KJ, Burrage PM.
Stability Switching in Lotka-Volterra and Ricker-Type Predator-Prey Systems with Arbitrary Step Size. *Axioms*. 2023; 12(4):390.
https://doi.org/10.3390/axioms12040390

**Chicago/Turabian Style**

Kekulthotuwage Don, Shamika, Kevin Burrage, Kate J. Helmstedt, and Pamela M. Burrage.
2023. "Stability Switching in Lotka-Volterra and Ricker-Type Predator-Prey Systems with Arbitrary Step Size" *Axioms* 12, no. 4: 390.
https://doi.org/10.3390/axioms12040390