# Directionally Correlated Movement Can Drive Qualitative Changes in Emergent Population Distribution Patterns

## Abstract

**:**

## 1. Introduction

## 2. The Modelling Framework and General Results

**Stable.**All eigenvalues have negative real part: Re$\left({\lambda}_{i}\left(\kappa \right)\right)<0$ for all $i\in \{1,\cdots ,N\}$, $\kappa >0$,**Turing instability.**The dominant eigenvalue (i.e., the one with the largest real part) is positive and real, i.e., ${\mathrm{argmax}}_{{\lambda}_{i}\left(\kappa \right)}\left[\mathrm{Re}\left({\lambda}_{i}\left(\kappa \right)\right)\right]\in {\mathbb{R}}_{>0}$,**Turing-Hopf instability.**The dominant eigenvalue is not real but has positive real part, i.e., ${\mathrm{argmax}}_{{\lambda}_{i}\left(\kappa \right)}\left[\mathrm{Re}\left({\lambda}_{i}\left(\kappa \right)\right)\right]\in \{z\in \mathbb{C}:\mathrm{Re}\left(z\right)>0,z\notin \mathbb{R}\}$.

**Theorem**

**1.**

- 1
- If ${\lambda}_{i}\left(\kappa \right)\in \mathbb{R}$ for all $i,\kappa $, then Re$\left({\sigma}_{i}^{\pm}(\kappa ,T)\right)<0$ for all $i,\kappa ,T$, so the system stays in the Stable regime for all $T>0$.
- 2
- If there exist i and κ such that ${\lambda}_{i}\left(\kappa \right)\notin \mathbb{R}$ then there exists some ${T}_{\ast}>0$ such that for all $T>{T}_{\ast}$, there is a Turing-Hopf instability. In other words, for this value of i and κ, ${\mathit{argmax}}_{{\sigma}_{i}(\kappa ,T)}[\mathit{Re}\left({\sigma}_{i}(\kappa ,T)\right]\in \{z\in \mathbb{C}:\mathit{Re}\left(z\right)>0,z\notin \mathbb{R}\}$ for all $T>{T}_{\ast}$. Furthermore, ${T}_{\ast}$ is the minimum $T>0$ such that there exist $i,\kappa $ with Re$\left(\sqrt{1+4T{\kappa}^{2}{\lambda}_{i}\left(\kappa \right)}\right)>1$.

**Proof.**

**Theorem**

**2.**

- 1
- If Re$\left(\sqrt{1+4{\kappa}^{2}T{\lambda}_{j}\left(\kappa \right)}\right)\le \sqrt{1+4{\kappa}^{2}T{\lambda}_{i}\left(\kappa \right)}$ for all j then there is a Turing instability at wavenumber κ and persistence time T.
- 2
- If there is some j such that Re$\left(\sqrt{1+4{\kappa}^{2}T{\lambda}_{j}\left(\kappa \right)}\right)>\sqrt{1+4{\kappa}^{2}T{\lambda}_{i}\left(\kappa \right)}$ then there is a Turing-Hopf instability at wavenumber κ and persistence time T. Let ${T}_{\ast}$ be the minimum $T>0$ such that Re$\left(\sqrt{1+4{\kappa}^{2}T{\lambda}_{j}\left(\kappa \right)}\right)>\sqrt{1+4{\kappa}^{2}T{\lambda}_{i}\left(\kappa \right)}$ for some j. Then there is a Turing-Hopf instability for all $T>{T}_{\ast}$.

**Proof.**

**Theorem**

**3.**

**Proof.**

## 3. The Case of Two Interacting Populations ($\mathit{N}=\mathbf{2}$)

## 4. Discussion

## Acknowledgments

## Conflicts of Interest

## References

- Johnson, C.J.; Seip, D.R.; Boyce, M.S. A quantitative approach to conservation planning: using resource selection functions to map the distribution of mountain caribou at multiple spatial scales. J. Appl. Ecol.
**2004**, 41, 238–251. [Google Scholar] [CrossRef] - Ferrier, S.; Guisan, A. Spatial modelling of biodiversity at the community level. J. Appl. Ecol.
**2006**, 43, 393–404. [Google Scholar] [CrossRef] - Peterson, A.T. Predicting the geography of species invasions via ecological niche modeling. Q. Rev. Biol.
**2003**, 78, 419–433. [Google Scholar] [CrossRef] [PubMed] - Lewis, M.A.; Petrovskii, S.V.; Potts, J.R. The Mathematics Behind Biological Invasions; Springer: Berlin, Germany, 2016; Volume 44. [Google Scholar]
- 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] [Green Version] - Börger, L. Stuck in motion? Reconnecting questions and tools in movement ecology. J. Anim. Ecol.
**2016**, 85, 5–10. [Google Scholar] [CrossRef] [PubMed] - Hays, G.C.; Ferreira, L.C.; Sequeira, A.M.; Meekan, M.G.; Duarte, C.M.; Bailey, H.; Bailleul, F.; Bowen, W.D.; Caley, M.J.; Costa, D.P.; et al. Key questions in marine megafauna movement ecology. Trends Ecol. Evol.
**2016**, 31, 463–475. [Google Scholar] [CrossRef] [PubMed] - Mueller, T.; Fagan, W.F. Search and navigation in dynamic environments–from individual behaviors to population distributions. Oikos
**2008**, 117, 654–664. [Google Scholar] [CrossRef] - Lewis, M.A.; Maini, P.K.; Petrovskii, S.V. Dispersal, Individual Movement and Spatial Ecology. In Lecture Notes in Mathematics (Mathematics Bioscience Series); Springer: Berlin/Heidelberg, Germany, 2013; Volume 2071. [Google Scholar] [CrossRef]
- Moorcroft, P.R.; Lewis, M.A.; Crabtree, R.L. Mechanistic home range models capture spatial patterns and dynamics of coyote territories in Yellowstone. Proc. R. Soc. B
**2006**, 273, 1651–1659. [Google Scholar] [CrossRef] [Green Version] - Potts, J.R.; Mokross, K.; Lewis, M.A. A unifying framework for quantifying the nature of animal interactions. J. R. Soc. Interface
**2014**, 11, 20140333. [Google Scholar] [CrossRef] [Green Version] - Potts, J.R.; Lewis, M.A. Spatial Memory and Taxis-Driven Pattern Formation in Model Ecosystems. Bull. Math. Biol.
**2019**. [Google Scholar] [CrossRef] - Shigesada, N.; Kawasaki, K.; Teramoto, E. Spatial segregation of interacting species. J. Theor. Biol.
**1979**, 79, 83–99. [Google Scholar] [CrossRef] - Durrett, R.; Levin, S. The importance of being discrete (and spatial). Theor. Pop. Biol.
**1994**, 46, 363–394. [Google Scholar] [CrossRef] - Sherratt, J.A.; Lewis, M.A.; Fowler, A.C. Ecological chaos in the wake of invasion. Proc. Natl. Acad. Sci. USA
**1995**, 92, 2524–2528. [Google Scholar] [CrossRef] [PubMed] - Lee, J.; Hillen, T.; Lewis, M. Pattern formation in prey-taxis systems. J. Biol. Dyn.
**2009**, 3, 551–573. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Gambino, G.; Lombardo, M.C.; Sammartino, M. A velocity–diffusion method for a Lotka–Volterra system with nonlinear cross and self-diffusion. Appl. Numer. Math.
**2009**, 59, 1059–1074. [Google Scholar] [CrossRef] - Kareiva, P.; Shigesada, N. Analyzing insect movement as a correlated random walk. Oecologia
**1983**, 56, 234–238. [Google Scholar] [CrossRef] [PubMed] - Codling, E.A.; Plank, M.J.; Benhamou, S. Random walk models in biology. J. R. Soc. Interface
**2008**, 5, 813–834. [Google Scholar] [CrossRef] - Wilson, R.P.; Griffiths, I.W.; Legg, P.A.; Friswell, M.I.; Bidder, O.R.; Halsey, L.G.; Lambertucci, S.A.; Shepard, E.L.C. Turn costs change the value of animal search paths. Ecol. Lett.
**2013**, 16, 1145–1150. [Google Scholar] [CrossRef] - Patlak, C.S. Random walk with persistence and external bias. Bull. Math. Biophys.
**1953**, 15, 311–338. [Google Scholar] [CrossRef] - Turchin, P. Translating foraging movements in heterogeneous environments into the spatial distribution of foragers. Ecology
**1991**, 72, 1253–1266. [Google Scholar] [CrossRef] - Giuggioli, L.; Potts, J.; Harris, S. Predicting oscillatory dynamics in the movement of territorial animals. J. R. Soc. Interface
**2012**, 9, 1529–1543. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Kac, M. A stochastic model related to the telegrapher’s equation. Rocky Mt. J. Math.
**1956**, 4, 497–509. [Google Scholar] [CrossRef] - Weiss, G.H. Some applications of persistent random walks and the telegrapher’s equation. Physica A
**2002**, 311, 381–410. [Google Scholar] [CrossRef] - Murray, J.D. Mathematical Biology II: Spatial Models and Biomedical Applications; Springer: New York, NY, USA, 2003. [Google Scholar]
- Turing, A.M. The chemical basis of morphogenesis. Philos. Trans. R. Soc. Lond. B
**1952**, 237, 37–72. [Google Scholar] - Painter, K.; Bloomfield, J.; Sherratt, J.; Gerisch, A. A nonlocal model for contact attraction and repulsion in heterogeneous cell populations. Bull. Math. Biol.
**2015**, 77, 1132–1165. [Google Scholar] [CrossRef] [PubMed] - Lewis, M.; Moorcroft, P. Mechanistic Home Range Analysis; Princeton University Press: Princeton, NJ, USA, 2006. [Google Scholar]
- Giuggioli, L.; Potts, J.R.; Rubenstein, D.I.; Levin, S.A. Stigmergy, collective actions, and animal social spacing. Proc. Natl. Acad. Sci. USA
**2013**, 110, 16904–16909. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Fagan, W.F.; Lewis, M.A.; Auger-Méthé, M.; Avgar, T.; Benhamou, S.; Breed, G.; LaDage, L.; Schlägel, U.E.; Tang, W.w.; Papastamatiou, Y.P.; et al. Spatial memory and animal movement. Ecol. Lett.
**2013**, 16, 1316–1329. [Google Scholar] [CrossRef] [PubMed] - Potts, J.R.; Lewis, M.A. How memory of direct animal interactions can lead to territorial pattern formation. J. R. Soc. Interface
**2016**, 13, 20160059. [Google Scholar] [CrossRef] - Zeigler, S.L.; Fagan, W.F. Transient windows for connectivity in a changing world. Mov. Ecol.
**2014**, 2, 1. [Google Scholar] [CrossRef] - Thurfjell, H.; Ciuti, S.; Boyce, M.S. Applications of step-selection functions in ecology and conservation. Mov. Ecol.
**2014**, 2, 4. [Google Scholar] [CrossRef] - Alt, W. Degenerate diffusion equations with drift functionals modelling aggregation. Nonlinear Anal. Theory Methods Appl.
**1985**, 9, 811–836. [Google Scholar] [CrossRef] - Stevens, A.; Othmer, H.G. Aggregation, blowup, and collapse: The ABC’s of taxis in reinforced random walks. SIAM J. Math. Appl.
**1997**, 57, 1044–1081. [Google Scholar] [CrossRef] - Mogilner, A.; Edelstein-Keshet, L. A non-local model for a swarm. J. Math. Biol.
**1999**, 38, 534–570. [Google Scholar] [CrossRef] - Topaz, C.M.; Bertozzi, A.L.; Lewis, M.A. A nonlocal continuum model for biological aggregation. Bull. Math. Biol.
**2006**, 68, 1601. [Google Scholar] [CrossRef] [PubMed] - Burger, M.; Francesco, M.D.; Fagioli, S.; Stevens, A. Sorting phenomena in a mathematical model for two mutually attracting/repelling species. SIAM J. Math. Appl.
**2018**, 50, 3210–3250. [Google Scholar] [CrossRef] - Wolansky, G. Multi-components chemotactic system in the absence of conflicts. Eur. J. Appl. Math.
**2002**, 13, 641–661. [Google Scholar] [CrossRef] - Horstmann, D. Generalizing the Keller–Segel model: Lyapunov functionals, steady state analysis, and blow-up results for multi-species chemotaxis models in the presence of attraction and repulsion between competitive interacting species. J. Nonlinear Sci.
**2011**, 21, 231–270. [Google Scholar] [CrossRef] - Hall, R. Amoeboid movement as a correlated walk. J. Math. Biol.
**1977**, 4, 327–335. [Google Scholar] [CrossRef] - Potdar, A.A.; Jeon, J.; Weaver, A.M.; Quaranta, V.; Cummings, P.T. Human mammary epithelial cells exhibit a bimodal correlated random walk pattern. PLoS ONE
**2010**, 5, e9636. [Google Scholar] [CrossRef] - Wadkin, L.; Orozco-Fuentes, S.; Neganova, I.; Swan, G.; Laude, A.; Lako, M.; Shukurov, A.; Parker, N. Correlated random walks of human embryonic stem cells in vitro. Phys. Biol.
**2018**, 15, 056006. [Google Scholar] [CrossRef] - Hillen, T. A Turing model with correlated random walk. J. Math. Biol.
**1996**, 35, 49–72. [Google Scholar] [CrossRef] - Chen, L.; Jüngel, A. Analysis of a multidimensional parabolic population model with strong cross-diffusion. SIAM J. Math. Anal.
**2004**, 36, 301–322. [Google Scholar] [CrossRef] - Zamponi, N.; Jüngel, A. Analysis of degenerate cross-diffusion population models with volume filling. In Annales de l’Institut Henri Poincare (C) Non Linear Analysis; Elsevier: Amsterdam, The Netherlands, 2017; Volume 34, pp. 1–29. [Google Scholar]
- Hernandez-Martinez, E.; Puebla, H.; Perez-Munoz, T.; Gonzalez-Brambila, M.; Velasco-Hernandez, J.X. Spatiotemporal Dynamics of Telegraph Reaction-Diffusion Predator-prey Models. In BIOMAT 2012; World Scientific: Singapore, 2013; pp. 268–281. [Google Scholar]

**Figure 1.**

**Critical value of T for pattern formation.**In Panel (

**a**), Equation (19) is plotted for $\gamma =0.2$, $\sigma =0.05$, and various values of T. Where $f\left(\kappa \right)>1$, there is a Turing-Hopf instability. In Panels (

**b**,

**c**), the value, ${T}_{\ast}$, above which there is a Turing-Hopf instability for some $\kappa $ and below which there is not, is plotted for various values of $\sigma $ and $\gamma $.

**Figure 2.**

**Example numerics for $N=2$.**Numerical solutions of Equation (1) for $N=2$, ${d}_{2}=1$, ${\gamma}_{12}=1$, ${\gamma}_{21}=-1$, and $\mathcal{K}\left(x\right)$ as in Equation (14) with $\sigma =0.1$. Both examples have random continuous initial conditions, but forced to be symmetric about $x=0.5$ to satisfy periodic boundary conditions. In Panel (

**a**), $T=0$, and the initial perturbation of the constant steady state decays to the solution ${u}_{1}(x,\infty )=1$. In Panel (

**b**), $T=1$. Here, the population ${u}_{1}$ moves across the landscape, not settling to a constant distribution.

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Potts, J.R.
Directionally Correlated Movement Can Drive Qualitative Changes in Emergent Population Distribution Patterns. *Mathematics* **2019**, *7*, 640.
https://doi.org/10.3390/math7070640

**AMA Style**

Potts JR.
Directionally Correlated Movement Can Drive Qualitative Changes in Emergent Population Distribution Patterns. *Mathematics*. 2019; 7(7):640.
https://doi.org/10.3390/math7070640

**Chicago/Turabian Style**

Potts, Jonathan R.
2019. "Directionally Correlated Movement Can Drive Qualitative Changes in Emergent Population Distribution Patterns" *Mathematics* 7, no. 7: 640.
https://doi.org/10.3390/math7070640