# Taxis-Driven Pattern Formation in Tri-Trophic Food Chain Model with Omnivory

## Abstract

**:**

## 1. Introduction

## 2. Model Formulation and Preliminary Analysis

**Proposition 1.**

- There exist equilibria ${E}_{0}$ and ${E}_{1}$ for any parameters.
- There exists equilibrium ${E}_{2}$ if ${\mu}_{1}>{m}_{1}(1+{\gamma}_{1})$.
- There exists equilibrium ${E}_{3}$ if ${\mu}_{2}>{m}_{2}(1+{\gamma}_{2})$.
- There exists equilibrium ${E}^{\ast}$ if ${\mu}_{1}>{m}_{1}$ and Equation (9) has a positive root ${u}^{\ast}$ such that the following conditions are satisfied:$${u}^{\ast}>\frac{{\gamma}_{1}{m}_{1}}{{\mu}_{1}-{m}_{1}},\phantom{\rule{1.em}{0ex}}{v}^{\ast}<\frac{{u}^{\ast}({\mu}_{1}-{m}_{1})-{\gamma}_{1}{m}_{1}}{{\delta}_{1}({\gamma}_{1}+{u}^{\ast})}.$$

**Proof.**

## 3. Classical Bifurcation Analysis

#### 3.1. Linear Analysis

#### 3.2. Existence of Nonconstant Positive Steady States

**Theorem 1.**

**Proof.**

## 4. Pattern Formation

#### 4.1. Conditions for Turing Instability

**Theorem 2.**

- ${G}_{1}(\eta -{\eta}_{1})<0$ and $\chi >{\chi}_{T1}$;
- ${G}_{1}(\eta -{\eta}_{1})>0$, $\chi >{\chi}_{T2}$, and there exist ${\chi}_{1}>{\overline{\chi}}_{1}$, $\eta >{\eta}_{2}$ such that ${G}_{3}>0$.

**Proof.**

#### 4.2. Conditions for Wave Instability

**Theorem 3.**

- ${G}_{4}(\eta -{\eta}_{3})<0$ and $\chi >{\chi}_{W1}$;
- ${G}_{4}(\eta -{\eta}_{3})>0$, $\chi >{\chi}_{W2}$, and there exist ${\chi}_{1}>{\tilde{\chi}}_{1}$ and $\eta <{\eta}_{4}$ such that ${G}_{5}>0$.

**Proof.**

## 5. Numerical Simulations

#### 5.1. Numerical Scheme

#### 5.2. Spatiotemporal Patterns

## 6. Discussion and Conclusions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Nathan, R.; Getz, W.M.; Revilla, E.; Holyoak, M.; Kadmon, R.; Saltz, D.; Smouse, P.E. A movement ecology paradigm for unifying organismal movement research. Proc. Natl. Acad. Sci. USA
**2008**, 105, 19052–19059. [Google Scholar] [CrossRef] [PubMed] - Quévreux, P.; Pigeault, R.; Loreau, M. Predator avoidance and foraging for food shape synchrony and response to perturbations in trophic metacommunities. J. Theor. Biol.
**2021**, 528, 110836. [Google Scholar] [CrossRef] [PubMed] - Holmes, E.E.; Lewis, M.A.; Banks, J.E.; Veit, R.R. Partial Differential Equations in Ecology: Spatial Interactions and Population Dynamics. Ecology
**1994**, 75, 17–29. [Google Scholar] [CrossRef] - Skellam, J.G. Random Dispersal in Theoretical Populations. Biometrika
**1951**, 38, 196–218. [Google Scholar] [CrossRef] [PubMed] - Kareiva, P.; Odell, G. Swarms of predators exhibit “preytaxis” if individual predators use area restricted search. Am. Nat.
**1987**, 130, 233–270. [Google Scholar] [CrossRef] - Berezovskaya, F.S.; Karev, G.P. Bifurcations of travelling waves in population taxis models. Phys. Uspekhi
**1999**, 42, 917–929. [Google Scholar] [CrossRef] - Turchin, P. Quantitative Analysis of Movement: Measuring and Modeling Population Redistribution in Animals and Plants; Sinauer Associates: Sunderland, MA, USA, 1998. [Google Scholar]
- Wang, Q.; Song, Y.; Shao, L. Nonconstant positive steady states and pattern formation of 1d prey-taxis systems. J. Nonlinear Sci.
**2017**, 27, 71–97. [Google Scholar] [CrossRef] - Wang, X.; Wang, W.; Zhang, G. Global bifurcation of solutions for a predator–prey model with prey-taxis. Math. Methods Appl. Sci.
**2015**, 38, 431–443. [Google Scholar] [CrossRef] - Lee, J.M.; Hillen, T.; Lewis, M.A. Pattern formation in prey-taxis systems. J. Biol. Dyn.
**2009**, 3, 551–573. [Google Scholar] [CrossRef] - Tyutyunov, Y.V.; Titova, L.I.; Senina, I.N. Prey-taxis destabilizes homogeneous stationary state in spatial Gause–Kolmogorov-type model for predator–prey system. Ecol. Complex.
**2017**, 31, 170–180. [Google Scholar] [CrossRef] - Pang, P.Y.H.; Wang, M. Strategy and stationary pattern in a three-species predator–prey model. J. Differ. Equ.
**2004**, 200, 245–273. [Google Scholar] [CrossRef] - Wang, K.; Wang, Q.; Yu, F. Stationary and time-periodic patterns of two-predator and one-prey systems with prey-taxis. Discret. Contin. Dyn. Syst.
**2017**, 37, 505–543. [Google Scholar] [CrossRef] - Hamilton, W.D. Geometry of the selfish herd. J. Theor. Biol.
**1971**, 31, 295–311. [Google Scholar] [CrossRef] - Xiao, D.; Ruan, S. Codimension two bifurcations in a predator–prey system with group defense. Int. J. Bifurc. Chaos
**2001**, 11, 2123–2131. [Google Scholar] [CrossRef] - Hsu, S.B.; Ruan, S.; Yang, T.H. Analysis of three species Lotka–Volterra food web models with omnivory. J. Math. Anal. Appl.
**2015**, 426, 659–687. [Google Scholar] [CrossRef] - Kumari, S.; Upadhyay, R.K. Dynamics comparison between non-spatial and spatial systems of the plankton–fish interaction model. Nonlinear Dyn.
**2020**, 99, 2479–2503. [Google Scholar] [CrossRef] - Mortoja, S.G.; Panja, P.; Mondal, S.K. Stability Analysis of Plankton–Fish Dynamics with Cannibalism Effect and Proportionate Harvesting on Fish. Mathematics
**2023**, 11, 3011. [Google Scholar] [CrossRef] - Giricheva, E. Stability and bifurcation analysis of a tri-trophic food chain model with intraguild predation. Int. J. Biomath.
**2023**, 16, 2250073. [Google Scholar] [CrossRef] - Wu, C.J.; Chiang, K.P.; Liu, H. Diel feeding pattern and prey selection of mesozooplankton on microplankton community. J. Exp. Mar. Biol. Ecol.
**2010**, 390, 134–142. [Google Scholar] [CrossRef] - Fuest, M. Global Solutions near Homogeneous Steady States in a Multidimensional Population Model with Both Predator- and Prey-Taxis. SIAM J. Math. Anal.
**2020**, 52, 5865–5891. [Google Scholar] [CrossRef] - Wang, J.F.; Wu, S.N.; Shi, J.P. Pattern formation in diffusive predator-prey systems with predator-taxis and prey-taxis. Discret. Contin. Dyn. Syst. Ser. B
**2021**, 26, 1273–1289. [Google Scholar] [CrossRef] - Guo, X.; Wang, J. Dynamics and pattern formations in diffusive predator-prey models with two prey-taxis. Math. Methods Appl. Sci.
**2019**, 42, 4197–4212. [Google Scholar] [CrossRef] - Han, R.; Röst, G. Stationary and oscillatory patterns of a food chain model with diffusion and predator-taxis. Math. Methods Appl. Sci.
**2023**, 46, 9652–9675. [Google Scholar] [CrossRef] - Malchow, H.; Petrovskii, S.V.; Venturino, E. Spatiotemporal Patterns in Ecology and Epidemiology: Theory, Models, and Simulation; CRC Press: Boca Raton, FL, USA, 2007. [Google Scholar]
- Crandall, M.G.; Rabinowitz, P.H. Bifurcation from simple eigenvalues. J. Funct. Anal.
**1971**, 8, 321–340. [Google Scholar] [CrossRef] - Shi, J.; Wang, X. On global bifurcation for quasilinear elliptic systems on bounded domains. J. Differ. Equ.
**2009**, 246, 2788–2812. [Google Scholar] [CrossRef] - Gerisch, A.; Griffiths, D.F.; Weiner, R.; Chaplain, M.A. A positive splitting method for mixed hyperbolic-parabolic systems. Numer. Methods Partial. Differ. Equ. Int. J.
**2001**, 17, 152–168. [Google Scholar] [CrossRef] - Hundsdorfer, W.H.; Verwer, J.G.; Hundsdorfer, W.H. Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations; Springer: Berlin/Heidelberg, Germany, 2003; Volume 33. [Google Scholar]
- Orrock, J.L.; Preisser, E.L.; Grabowski, J.H.; Trussell, G.C. The cost of safety: Refuges increase the impact of predation risk in aquatic systems. Ecology
**2013**, 94, 573–579. [Google Scholar] [CrossRef] - Abraham, E.R. The generation of plankton patchiness by turbulent stirring. Nature
**1998**, 391, 577–580. [Google Scholar] [CrossRef] - Semplice, M.; Venturino, E. Travelling waves in plankton dynamics. Math. Model. Nat. Phenom.
**2013**, 8, 64–79. [Google Scholar] [CrossRef] - Ivanitsky, G.R.; Medvinskii, A.B.; Tsyganov, M.A. From the dynamics of population autowaves generated by living cells to neuroinformatics. Phys. Uspekhi
**1994**, 37, 961–989. [Google Scholar] [CrossRef] - Morozov, A.; Petrovskii, S.; Li, B.L. Spatiotemporal complexity of patchy invasion in a predator-prey system with the Allee effect. J. Theor. Biol.
**2006**, 238, 18–35. [Google Scholar] [CrossRef] [PubMed] - Bate, A.M.; Hilker, F.M. Preytaxis and travelling waves in an eco-epidemiological model. Bull. Math. Biol.
**2019**, 81, 995–1030. [Google Scholar] [CrossRef] [PubMed] - Yang, L.A. Pattern formation arising from interactions between Turing and wave instabilities. J. Chem. Phys.
**2002**, 117, 7259–7265. [Google Scholar] [CrossRef] - Kaminaga, A.; Vanag, V.K.; Epstein, I.R. A reaction–diffusion memory device. Angew. Chem.
**2006**, 45, 3087–3089. [Google Scholar] [CrossRef]

**Figure 1.**Stability/instability domains: ${\Omega}_{1}$—stability area, ${\Omega}_{2}$—Turing instability, ${\Omega}_{3}$—wave instability, and ${\Omega}_{4}$—Turing and wave instabilities.

**Figure 2.**Stationary and oscillatory patterns for $\eta =0$ and $\chi =2$. Top panel: ${\chi}_{1}=0$; middle panel: ${\chi}_{1}=0.1$; bottom panel: ${\chi}_{1}=2$.

**Figure 3.**Stationary and oscillatory patterns for $\chi =2$, ${\chi}_{1}=0.1$. Top panel: ${\chi}_{2}=0.1$; bottom panel: ${\chi}_{2}=1$.

**Figure 4.**System dynamics for $\chi =2$, ${\chi}_{1}=0.1$. Top panel: ${\chi}_{2}=2$; bottom panel: ${\chi}_{2}=4$.

**Figure 5.**Spatially homogeneous solutions to System (3) with an unstable spatially homogeneous equilibrium as the initial state: (

**a**) ${d}_{i}=0$, $\chi ={\chi}_{1}={\chi}_{2}=0$; (

**b**) ${d}_{i}={10}^{-4}$, $\chi =0.5$, ${\chi}_{1}={\chi}_{2}=0$.

**Figure 6.**Spatially nonhomogeneous solutions to System (3) with an unstable spatially homogeneous equilibrium as the initial state with ${\chi}_{1}=0.1$: (

**a**) $\chi ={\chi}_{2}=0$; (

**b**) $\chi =0.5$, ${\chi}_{2}=0.05$; (

**c**) $\chi =0.5$, ${\chi}_{2}=0.5$.

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2024 by the author. 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**

Giricheva, E.
Taxis-Driven Pattern Formation in Tri-Trophic Food Chain Model with Omnivory. *Mathematics* **2024**, *12*, 290.
https://doi.org/10.3390/math12020290

**AMA Style**

Giricheva E.
Taxis-Driven Pattern Formation in Tri-Trophic Food Chain Model with Omnivory. *Mathematics*. 2024; 12(2):290.
https://doi.org/10.3390/math12020290

**Chicago/Turabian Style**

Giricheva, Evgeniya.
2024. "Taxis-Driven Pattern Formation in Tri-Trophic Food Chain Model with Omnivory" *Mathematics* 12, no. 2: 290.
https://doi.org/10.3390/math12020290