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-{"title":"Mathematical Modeling in Ecology and Evolution","markdown":{"yaml":{"title":"Mathematical Modeling in Ecology and Evolution","description":"This workshop will introduce participants to the logic behind modeling in biology, focusing on developing equations, finding equilibria, analyzing stability, and running simulations.Techniques will be illustrated with the software tools, Mathematica and Maxima. This workshop was held in two parts: January 14 and January 16, 2020.\n","categories":["Technical","EN"],"author":[{"name":"Sarah P. Otto","affiliation":"University of British Columbia"}],"base_url":"https://bios2.github.io/","bibliography":"citations.bib","image":"StabilityPictures.jpg","date":"2020-01-14","toc":true,"number-sections":true,"number-depth":1},"headingText":"Content","containsRefs":false,"markdown":"\n\nIn this workshop, I introduce various modelling techniques, using mostly ecological and evolutionary examples, with a focus on how computer software programs can help biologists analyze such models.\n\n\n**Part 1:** Classic one-variable models in ecology and evolution\\\n**Part 2:** Equilibria and their stability\\\n**Part 3:** Beyond equilibria\\\n**Part 4:** Example of building a model from scratch\\\n**Part 5:** Extending to models with more than one variable\\\n**Part 6:** Another example of building a model from scratch\n\n### Software\n\nIn my research, I primarily use Mathematica, which is a powerful software package to organize and conduct analytical modelling, but it is not free (at UBC, we have some licenses available). I will also show some example code and provide translation of most of what I present in a free software package called Maxima.\n\n#### Mathematica installation\n\nThere is a free trial version that you can use for 15 days, if you don't have a copy (click [here](http://www.wolfram.com/mathematica/trial/) to access), or you can buy a [student version](https://www.wolfram.com/mathematica/pricing/students/) online. If you want to make sure that all is working, copy the code below, put your cursor over each of the following lines and press enter (on some computers, \"enter\" is a separate button, on others, press \"shift\" and \"return\" at the same time):\n\n``` mathematica\nD[x^3,x]\nListPlot[Table[x, {x,1,10}],Joined->True]\nRSolve[{x[t+1]\\[Equal]A x[t],x[0]\\[Equal]x0},x[t],t]\nPDF[NormalDistribution[0,1],x]\n```\n\nYou should see (a) $3x^2$, (b) a plot of a line, (c) ${{x[t]->A^t x0}}$, and (d) $\\frac{e^\\frac{-x^2}{2}}{\\sqrt{2\\pi }}$.\n\n#### Maxima installation:\n\nOn a Mac, install using the instructions [here](https://themaximalist.org/about/my-mac-os-installation/). For other file systems, download [here](https://sourceforge.net/projects/maxima/files/).\n\n#### Maxima testing\n\nWhen you first open Maxima, it will give you a choice of GUIs, chose wxMaxima. Once wxMaxima is launched type this command and hit return to see if it answers 4:\n\n``` maxima\n2+2;\n```\n\nIf it doesn't, then scan the installation document for the error that you run into.\n\nIf it does return 4, then type in and enter these commands:\n\n``` maxima\ndiff(x^3, x);\n\nwxplot2d (3*x, [x, 0, 2*%pi]);\n\nload(\"solve_rec\")$\nsolve_rec(x[t+1] = A*x[t], x[t], x[0]=x0);\n\nload(\"distrib\")$\npdf_normal(x,0,1);\n```\n\nYou should see (a) $3x^2$, (b) a plot of a line, (c) ${{x[t]->A^t x0}}$, and (d) $\\frac{e^\\frac{-x^2}{2}}{\\sqrt{2\\pi }}$.\n\n### Material\n\n|                            Mathematica                             |                             Maxima                              |                      PDF                      |\n|:----------------------:|:----------------------:|:----------------------:|\n|           [Notebook](files/BIOS2_WorkshopMathematica.nb)           |           [Notebook](files/Bios2_WorkshopMaxima.wxmx)           |                 Embeded below                 |\n| [Hints and solutions](files/BIOS2_WorkshopMathematicaSOLUTIONS.nb) | [Hints and solutions](files/Bios2_WorkshopMaximaSOLUTIONS.wxmx) |                                               |\n|                                                                    |                                                                 |        [Homework](files/Homework1.pdf)        |\n|            [Homework answers](files/HomeworkAnswers.nb)            |                                                                 | [Homework answers](files/HomeworkAnswers.pdf) |\n|                 [Guide](files/MathematicaGuide.nb)                 |             [Guide](files/MaximaQuickReference.wxm)             |                                               |\n\n#### Follow along PDF\n\nThis PDF was generated from the Mathematica notebook linked above. It doesn't include dynamic plots, but it's a good alternative if you want to print out or have a quick reference at hand.\n\n<iframe src=\"files/otto1.nb.pdf\" width=\"100%\" height=\"600px\">\n\n</iframe>\n\n<br> <br>\n\n![Stability analysis of a recursion equation in a discrete-time model.](StabilityPictures.png)\n\n### Other resources\n\n-   *An Introduction to Mathematical Modeling in Ecology and Evolution* [@otto2007biologist].\n\n-   [Biomathematical modeling lecture notes](http://www.zoology.ubc.ca/~bio301/Bio301/Lectures.html).\n\n-   [Mathematica labs UBC](http://www.zoology.ubc.ca/biomath/labs.htm).\n\n### Thanks\n\nNiki Love and Gil Henriques did a great job of translating the code into wxMaxima, with limited help from me. Thanks, Niki and Gil!!\n"},"formats":{"html":{"execute":{"fig-width":7,"fig-height":5,"fig-format":"retina","fig-dpi":96,"df-print":"default","error":false,"eval":true,"cache":null,"freeze":true,"echo":true,"output":true,"warning":true,"include":true,"keep-md":false,"keep-ipynb":false,"ipynb":null,"enabled":null,"daemon":null,"daemon-restart":false,"debug":false,"ipynb-filters":[],"engine":"markdown"},"render":{"keep-tex":false,"keep-source":false,"keep-hidden":false,"prefer-html":false,"output-divs":true,"output-ext":"html","fig-align":"default","fig-pos":null,"fig-env":null,"code-fold":"none","code-overflow":"scroll","code-link":false,"code-line-numbers":false,"code-tools":false,"tbl-colwidths":"auto","merge-includes":true,"latex-auto-mk":true,"latex-auto-install":true,"latex-clean":true,"latex-max-runs":10,"latex-makeindex":"makeindex","latex-makeindex-opts":[],"latex-tlmgr-opts":[],"latex-input-paths":[],"latex-output-dir":null,"link-external-icon":false,"link-external-newwindow":false,"self-contained-math":false,"format-resources":[]},"pandoc":{"standalone":true,"wrap":"none","default-image-extension":"png","to":"html","css":["../../styles.css"],"toc":true,"number-sections":true,"output-file":"index.html"},"language":{},"metadata":{"lang":"en","fig-responsive":true,"quarto-version":"1.2.253","editor":"source","theme":"flatly","title-block-banner":true,"title":"Mathematical Modeling in Ecology and Evolution","description":"This workshop will introduce participants to the logic behind modeling in biology, focusing on developing equations, finding equilibria, analyzing stability, and running simulations.Techniques will be illustrated with the software tools, Mathematica and Maxima. This workshop was held in two parts: January 14 and January 16, 2020.\n","categories":["Technical","EN"],"author":[{"name":"Sarah P. Otto","affiliation":"University of British Columbia"}],"base_url":"https://bios2.github.io/","bibliography":["citations.bib"],"image":"StabilityPictures.jpg","date":"2020-01-14","number-depth":1},"extensions":{"book":{"multiFile":true}}}}}
+{"title":"Mathematical Modeling in Ecology and Evolution","markdown":{"yaml":{"title":"Mathematical Modeling in Ecology and Evolution","description":"This workshop will introduce participants to the logic behind modeling in biology, focusing on developing equations, finding equilibria, analyzing stability, and running simulations.Techniques will be illustrated with the software tools, Mathematica and Maxima. This workshop was held in two parts: January 14 and January 16, 2020.\n","categories":["Technical","EN"],"author":[{"name":"Sarah P. Otto","affiliation":"University of British Columbia"}],"base_url":"https://bios2.github.io/","bibliography":"citations.bib","image":"StabilityPictures.jpg","date":"2020-01-14","toc":true,"number-sections":true,"number-depth":1},"headingText":"Content","containsRefs":false,"markdown":"\n\nIn this workshop, I introduce various modelling techniques, using mostly ecological and evolutionary examples, with a focus on how computer software programs can help biologists analyze such models.\n\n\n**Part 1:** Classic one-variable models in ecology and evolution\\\n**Part 2:** Equilibria and their stability\\\n**Part 3:** Beyond equilibria\\\n**Part 4:** Example of building a model from scratch\\\n**Part 5:** Extending to models with more than one variable\\\n**Part 6:** Another example of building a model from scratch\n\n### Software\n\nIn my research, I primarily use Mathematica, which is a powerful software package to organize and conduct analytical modelling, but it is not free (at UBC, we have some licenses available). I will also show some example code and provide translation of most of what I present in a free software package called Maxima.\n\n#### Mathematica installation\n\nThere is a free trial version that you can use for 15 days, if you don't have a copy (click [here](http://www.wolfram.com/mathematica/trial/) to access), or you can buy a [student version](https://www.wolfram.com/mathematica/pricing/students/) online. If you want to make sure that all is working, copy the code below, put your cursor over each of the following lines and press enter (on some computers, \"enter\" is a separate button, on others, press \"shift\" and \"return\" at the same time):\n\n``` mathematica\nD[x^3,x]\nListPlot[Table[x, {x,1,10}],Joined->True]\nRSolve[{x[t+1]\\[Equal]A x[t],x[0]\\[Equal]x0},x[t],t]\nPDF[NormalDistribution[0,1],x]\n```\n\nYou should see (a) $3x^2$, (b) a plot of a line, (c) ${{x[t]->A^t x0}}$, and (d) $\\frac{e^\\frac{-x^2}{2}}{\\sqrt{2\\pi }}$.\n\n#### Maxima installation:\n\nOn a Mac, install using the instructions [here](https://themaximalist.org/about/my-mac-os-installation/). For other file systems, download [here](https://sourceforge.net/projects/maxima/files/).\n\n#### Maxima testing\n\nWhen you first open Maxima, it will give you a choice of GUIs, chose wxMaxima. Once wxMaxima is launched type this command and hit return to see if it answers 4:\n\n``` maxima\n2+2;\n```\n\nIf it doesn't, then scan the installation document for the error that you run into.\n\nIf it does return 4, then type in and enter these commands:\n\n``` maxima\ndiff(x^3, x);\n\nwxplot2d (3*x, [x, 0, 2*%pi]);\n\nload(\"solve_rec\")$\nsolve_rec(x[t+1] = A*x[t], x[t], x[0]=x0);\n\nload(\"distrib\")$\npdf_normal(x,0,1);\n```\n\nYou should see (a) $3x^2$, (b) a plot of a line, (c) ${{x[t]->A^t x0}}$, and (d) $\\frac{e^\\frac{-x^2}{2}}{\\sqrt{2\\pi }}$.\n\n### Material\n\n|                            Mathematica                             |                             Maxima                              |                      PDF                      |\n|:----------------------:|:----------------------:|:----------------------:|\n|           [Notebook](files/BIOS2_WorkshopMathematica.nb)           |           [Notebook](files/Bios2_WorkshopMaxima.wxmx)           |                 Embeded below                 |\n| [Hints and solutions](files/BIOS2_WorkshopMathematicaSOLUTIONS.nb) | [Hints and solutions](files/Bios2_WorkshopMaximaSOLUTIONS.wxmx) |                                               |\n|                                                                    |                                                                 |        [Homework](files/Homework1.pdf)        |\n|            [Homework answers](files/HomeworkAnswers.nb)            |                                                                 | [Homework answers](files/HomeworkAnswers.pdf) |\n|                 [Guide](files/MathematicaGuide.nb)                 |             [Guide](files/MaximaQuickReference.wxm)             |                                               |\n\n#### Follow along PDF\n\nThis PDF was generated from the Mathematica notebook linked above. It doesn't include dynamic plots, but it's a good alternative if you want to print out or have a quick reference at hand.\n\n<iframe src=\"files/otto1.nb.pdf\" width=\"100%\" height=\"600px\">\n\n</iframe>\n\n<br> <br>\n\n![Stability analysis of a recursion equation in a discrete-time model.](StabilityPictures.png)\n\n### Other resources\n\n-   *An Introduction to Mathematical Modeling in Ecology and Evolution* [@otto2007biologist].\n\n-   [Biomathematical modeling lecture notes](http://www.zoology.ubc.ca/~bio301/Bio301/Lectures.html).\n\n-   [Mathematica labs UBC](http://www.zoology.ubc.ca/biomath/labs.htm).\n\n### Thanks\n\nNiki Love and Gil Henriques did a great job of translating the code into wxMaxima, with limited help from me. Thanks, Niki and Gil!!\n"},"formats":{"html":{"execute":{"fig-width":7,"fig-height":5,"fig-format":"retina","fig-dpi":96,"df-print":"default","error":false,"eval":true,"cache":null,"freeze":true,"echo":true,"output":true,"warning":true,"include":true,"keep-md":false,"keep-ipynb":false,"ipynb":null,"enabled":null,"daemon":null,"daemon-restart":false,"debug":false,"ipynb-filters":[],"engine":"markdown"},"render":{"keep-tex":false,"keep-source":false,"keep-hidden":false,"prefer-html":false,"output-divs":true,"output-ext":"html","fig-align":"default","fig-pos":null,"fig-env":null,"code-fold":"none","code-overflow":"scroll","code-link":false,"code-line-numbers":false,"code-tools":false,"tbl-colwidths":"auto","merge-includes":true,"latex-auto-mk":true,"latex-auto-install":true,"latex-clean":true,"latex-max-runs":10,"latex-makeindex":"makeindex","latex-makeindex-opts":[],"latex-tlmgr-opts":[],"latex-input-paths":[],"latex-output-dir":null,"link-external-icon":false,"link-external-newwindow":false,"self-contained-math":false,"format-resources":[]},"pandoc":{"standalone":true,"wrap":"none","default-image-extension":"png","to":"html","css":["../../styles.css"],"toc":true,"number-sections":true,"output-file":"index.html"},"language":{},"metadata":{"lang":"en","fig-responsive":true,"quarto-version":"1.2.253","editor":"source","theme":"flatly","title-block-banner":true,"citation":{"container-title":"BIOS² Education Resources"},"title":"Mathematical Modeling in Ecology and Evolution","description":"This workshop will introduce participants to the logic behind modeling in biology, focusing on developing equations, finding equilibria, analyzing stability, and running simulations.Techniques will be illustrated with the software tools, Mathematica and Maxima. This workshop was held in two parts: January 14 and January 16, 2020.\n","categories":["Technical","EN"],"author":[{"name":"Sarah P. Otto","affiliation":"University of British Columbia"}],"base_url":"https://bios2.github.io/","bibliography":["citations.bib"],"image":"StabilityPictures.jpg","date":"2020-01-14","number-depth":1},"extensions":{"book":{"multiFile":true}}}}}