A Multispecies Cross-Diffusion Model for Territorial Development
We develop an agent-based model on a lattice to investigate territorial development motivated by markings such as graffiti, generalizing a previously-published model to account for <i>K</i> groups instead of two groups. We then analyze this model and present two novel variations. Our mod...
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| Format: | Article |
| Language: | English |
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MDPI AG
2021-06-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/9/12/1428 |
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| author | Abdulaziz Alsenafi Alethea B. T. Barbaro |
| author_facet | Abdulaziz Alsenafi Alethea B. T. Barbaro |
| author_sort | Abdulaziz Alsenafi |
| collection | DOAJ |
| description | We develop an agent-based model on a lattice to investigate territorial development motivated by markings such as graffiti, generalizing a previously-published model to account for <i>K</i> groups instead of two groups. We then analyze this model and present two novel variations. Our model assumes that agents’ movement is a biased random walk away from rival groups’ markings. All interactions between agents are indirect, mediated through the markings. We numerically demonstrate that in a system of three groups, the groups segregate in certain parameter regimes. Starting from the discrete model, we formally derive the continuum system of <inline-formula><math display="inline"><semantics><mrow><mn>2</mn><mi>K</mi></mrow></semantics></math></inline-formula> convection–diffusion equations for our model. These equations exhibit cross-diffusion due to the avoidance of the rival groups’ markings. Both through numerical simulations and through a linear stability analysis of the continuum system, we find that many of the same properties hold for the <i>K</i>-group model as for the two-group model. We then introduce two novel variations of the agent-based model, one corresponding to some groups being more timid than others, and the other corresponding to some groups being more threatening than others. These variations present different territorial patterns than those found in the original model. We derive corresponding systems of convection–diffusion equations for each of these variations, finding both numerically and through linear stability analysis that each variation exhibits a phase transition. |
| first_indexed | 2024-03-10T10:16:10Z |
| format | Article |
| id | doaj.art-ff338bb2be26422ab2cd08fcddffabf2 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T10:16:10Z |
| publishDate | 2021-06-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-ff338bb2be26422ab2cd08fcddffabf22023-11-22T00:49:17ZengMDPI AGMathematics2227-73902021-06-01912142810.3390/math9121428A Multispecies Cross-Diffusion Model for Territorial DevelopmentAbdulaziz Alsenafi0Alethea B. T. Barbaro1Department of Mathematics, Kuwait University, Kuwait City 12037, KuwaitDelft Institute of Applied Mathematics, Faculty of Electrical Engineering, Mathematics and Computer Science, Delft University of Technology, 2628 CD Delft, The NetherlandsWe develop an agent-based model on a lattice to investigate territorial development motivated by markings such as graffiti, generalizing a previously-published model to account for <i>K</i> groups instead of two groups. We then analyze this model and present two novel variations. Our model assumes that agents’ movement is a biased random walk away from rival groups’ markings. All interactions between agents are indirect, mediated through the markings. We numerically demonstrate that in a system of three groups, the groups segregate in certain parameter regimes. Starting from the discrete model, we formally derive the continuum system of <inline-formula><math display="inline"><semantics><mrow><mn>2</mn><mi>K</mi></mrow></semantics></math></inline-formula> convection–diffusion equations for our model. These equations exhibit cross-diffusion due to the avoidance of the rival groups’ markings. Both through numerical simulations and through a linear stability analysis of the continuum system, we find that many of the same properties hold for the <i>K</i>-group model as for the two-group model. We then introduce two novel variations of the agent-based model, one corresponding to some groups being more timid than others, and the other corresponding to some groups being more threatening than others. These variations present different territorial patterns than those found in the original model. We derive corresponding systems of convection–diffusion equations for each of these variations, finding both numerically and through linear stability analysis that each variation exhibits a phase transition.https://www.mdpi.com/2227-7390/9/12/1428agent-based modelphase transitioncross-diffusionmovement ecologysegregation modelpattern formation |
| spellingShingle | Abdulaziz Alsenafi Alethea B. T. Barbaro A Multispecies Cross-Diffusion Model for Territorial Development Mathematics agent-based model phase transition cross-diffusion movement ecology segregation model pattern formation |
| title | A Multispecies Cross-Diffusion Model for Territorial Development |
| title_full | A Multispecies Cross-Diffusion Model for Territorial Development |
| title_fullStr | A Multispecies Cross-Diffusion Model for Territorial Development |
| title_full_unstemmed | A Multispecies Cross-Diffusion Model for Territorial Development |
| title_short | A Multispecies Cross-Diffusion Model for Territorial Development |
| title_sort | multispecies cross diffusion model for territorial development |
| topic | agent-based model phase transition cross-diffusion movement ecology segregation model pattern formation |
| url | https://www.mdpi.com/2227-7390/9/12/1428 |
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