In gauravsk/ecoevoapps: Simulate Dynamics of Ecology/Evolution Models

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library(ecoevoapps)
library(deSolve)
library(tidyverse)
library(patchwork)
library(RColorBrewer)
library(knitr)
library(kableExtra)
knitr::opts_chunk$set(echo=FALSE)
theme_set(ecoevoapps::theme_apps())

Lotka-Volterra 竞争方程描述了两个彼此竞争食物、空间、及其他资源的物种，和两者之间的动态。该模型基于以下理念：当一个物种的族群数量增加时，它会开始受限于同物种个体之间的竞争（种内竞争），以及其他物种的竞争（种间竞争）。许多生物学的教科书及网站都有该模型的详细描述。

经典的Lotka-Volterra竞争模型有多种构建方法。其中的主要方法是用各物种的环境承载力($K_1$和$K_2$)和相对种间竞争强度（$\alpha$和$\beta$）来描述竞争。下文第一个选项卡中的模型即使用了这种构建。而Peter Chesson在他发表于2000年的经典论文"Mechanisms of Maintenance of Speices Diversity"中则提倡使用各物种的绝对竞争系数来描述Lotka-Volterra竞争方程。该模型构建可详见下文的第二个选项卡。

{.tabset}

使用环境承载力和相对种间竞争强度的模型

使用环境承载力,Lotka-Volterra竞争方程可表述为： $$\frac{dN_1}{dt} = r_1N_1\left(1 - \frac{N_1 + \alpha N_2}{K_1}\right)$$ $$\frac{dN_2}{dt} = r_2N_2\left(1 - \frac{N_2 + \beta N_1}{K_2}\right)$$

pars_vars <- c("$r_1$", 
               "$r_2$", 
               "$N_1$", 
               "$N_2$", 
               "$K_1$", 
               "$K_2$", 
               "$\\alpha$", 
               "$\\beta$")
descriptions <- c("物种1的自然生长率（intrinsic growth rate）",
                 "物种2的自然生长率",
                 "物种1的族群规模",
                 "物种2的族群规模",
                 "物种1的环境承载力",
                 "物种2的环境承载力",
                 "物种2对于物种1的相对人均效应（relative per capita effect）",
                 "物种1对于物种2的相对人均效应")
param_df <- data.frame(pars_vars, descriptions)
kable(x = param_df, format = "html", 
      col.names = c("参数/变量", "描述")) %>%
  kable_styling(full_width = FALSE, 
                bootstrap_options = c("striped", "hover", "condensed"),
                position = "center")

零增长等斜线 (求解 $N_2$): $$N_2 = - \frac{N_1}{\alpha} + \frac{K_1}{\alpha}$$ $$N_2 = -\beta N_1 + K_2$$

---

sidebarLayout(
  sidebarPanel(
    # Allow user to select which plots to display
    checkboxGroupInput(inputId = "plots_to_show",
                       label = "勾选以展示图表",
                       choices = c("Zero net growh isoclines (ZNGIs)" = "ZNGI")),

    hr(),

    # User-defined parameter values
    sliderInput(inputId = "r1", 
                label = HTML("r<sub>1</sub>: 物种1的自然增长率"),
                min = 0.01, max = 1, value = 0.2, step = 0.01),
    sliderInput(inputId = "r2", 
                label = HTML("r<sub>2</sub>: 物种2的自然增长率"),
                min = 0.01, max = 1, value = 0.5, step = 0.01),
    numericInput(inputId = "K1", 
                 label = HTML("K<sub>1</sub>: 物种1的环境承载力"),
                 min = 1, value = 300),
    numericInput(inputId = "K2", 
                 label = HTML("K<sub>2</sub>: 物种2的环境承载力"),
                 min = 1, value = 200),
    sliderInput(inputId = "alpha", 
                label = HTML("&alpha;: 物种2对物种1的相对影响"),
                min = 0.01, max = 2, value = 0.3, step = 0.01),
    sliderInput(inputId = "beta", 
                label = HTML("&beta;: 物种1对物种2的相对影响"),
                min = 0.01, max = 2, value = 0.2, step = 0.01),
    # User-defined initial values
    numericInput(inputId = "N1", 
                 label = "物种1的起始种群规模",
                 min = 1, value = 50),
    numericInput(inputId = "N2", 
                 label = "物种2的起始种群规模",
                 min = 1, value = 70),
    numericInput(inputId = "max_time", 
                 label = "模拟时长",
                 min = 1, value = 100)
  ),

  # Panel of plots
  mainPanel(
    renderPlot(N_vs_Time(), width = 600, height = 500),
      renderPlot(ZNGI(), width = 600, height = 500))
)

# Get user-defined parameters
init <- reactive({
  c(N1 = input$N1, N2 = input$N2)
})
time <- reactive({
  seq(from = 0, to = input$max_time, by = 0.1)
})
params <- reactive({
  c(r1 = input$r1, r2 = input$r2, K1 = input$K1, K2 = input$K2, 
    a = input$alpha, b = input$beta)
})



# Run lotka_volterra_competition function
lvcomp_out <- reactive({
  run_lvcomp_model(time = time(), init = init(), params = params()) %>% 
    data.frame()
})

# Plot N vs time for both species
N_vs_Time <- reactive({
  plot_lvcomp_time(lvcomp_out()) 
})

# Plot ZNGIs with population trajectories
ZNGI <- reactive({
  if ("ZNGI" %in% input$plots_to_show) {
    plot_lvcomp_portrait(lvcomp_out(), params = params())
  } 
})

# Make a list of plots to print based on user request
plot_list <- reactive({ 
  list(N_vs_Time(), ZNGI()) %>%
    discard(is.null)
  })

plots_to_print <- reactive({ 
  wrap_plots(plot_list(), ncol = 1)
  })

使用绝对竞争系数的模型

若不使用环境承载力，则Lotka-Volterra竞争方程可写为： $$\frac{dN_1}{dt} = r_1N_1\left(1 - \alpha_{11}N_1 - \alpha_{12}N_2\right)$$ $$\frac{dN_2}{dt} = r_2N_2\left(1 - \alpha_{22}N_2 - \alpha_{21}N_1\right)$$

pars_vars_wo_K <- c("$r_1$",
                    "$r_2$",
                    "$N_1$",
                    "$N_2$",
                    "$\\alpha_{11}$",
                    "$\\alpha_{12}$",
                    "$\\alpha_{22}$",
                    "$\\alpha_{21}$")
descriptions_wo_K <- c("物种1的自然增长率",
                       "物种2的自然增长",
                       "物种1的种群规模",
                       "物种2的种群规模",
                       "物种1对于自身的人均影响",
                       "物种2对于物种1的人均影响",
                       "物种2对于自身的人均影响",
                       "物种1对于物种2的人均影响")
param_df_wo_K <- data.frame(pars_vars_wo_K, descriptions_wo_K)
kable(x = param_df_wo_K, format = "html",
      col.names = c("参数/变量", "描述")) %>%
  kable_styling(full_width = FALSE,
                bootstrap_options = c("striped", "hover", "condensed"),
                position = "center")

零增长等斜线 (求解 $N_2$): $$N_2 = -\frac{\alpha_{11}}{\alpha_{12}}N_1 + \frac{1}{\alpha_{12}}$$ $$N_2 = -\frac{\alpha_{21}}{\alpha_{22}}N_1 + \frac{1}{\alpha_{22}}$$

---

sidebarLayout(
  sidebarPanel(
    # Allow users to select which plots to display
    checkboxGroupInput(inputId = "plots_to_show_wo_K",
                       label = "选择以展示图表",
                       choices = c("N vs. Time" = "N_vs_Time_wo_K",
                                   "Zero net growth isoclines (ZNGIs)" = "ZNGI_wo_K"),
                       selected = c("N_vs_Time_wo_K")),

    hr(),

    # User-defined parameter values
    sliderInput(inputId = "r1_wo_K",
                label = HTML("r<sub>1</sub>: 物种1的自然增长率"),
                min = 0.01, max = 1, value = 0.2, step = 0.01),
    sliderInput(inputId = "r2_wo_K",
                label = HTML("r<sub>2</sub>: 物种2的自然增长率"),
                min = 0.01, max = 1, value = 0.5, step = 0.01),

    # Allow user to choose input method for competition coefficients
    radioButtons(inputId = "alpha_input_wo_K",
                 label = "竞争系数的输入方式:",
                 choices = c("滑动滑块" = "slider", "手动输入" = "manual"),
                 selected = "slider"),

    # Conditional panel for slider input of competition coefficients
    conditionalPanel(
      condition = "input.alpha_input_wo_K == 'slider'",
      sliderInput(inputId = "alpha11_slider",
                  label = HTML("&alpha;<sub>11</sub>: 物种1对于自身的人均影响"),
                  min = 0.0001, max = 0.01, value = 0.0033, step = NULL),
      sliderInput(inputId = "alpha12_slider",
                  label = HTML("&alpha;<sub>12</sub>: 物种2对于物种1的人均影响"),
                  min = 0.0001, max = 0.01, value = 0.001, step = NULL),
      sliderInput(inputId = "alpha22_slider",
                  label = HTML("&alpha;<sub>22</sub>: 物种2对于自身的人均影响"),
                  min = 0.0001, max = 0.01, value = 0.005, step = NULL),
      sliderInput(inputId = "alpha21_slider",
                  label = HTML("&alpha;<sub>21</sub>: 物种1对于物种2的人均影响"),
                  min = 0.0001, max = 0.01, value = 0.001, step = NULL)
    ),

    # Conditional panel for manual input of competition coefficients
    conditionalPanel(
      condition = "input.alpha_input_wo_K == 'manual'",
      numericInput(inputId = "alpha11_manual",
                   label = HTML("&alpha;<sub>11</sub>: 物种1对于自身的人均影响"),
                   min = 0.0001, max = 1, value = 0.0033, step = 0.0001),
      numericInput(inputId = "alpha12_manual",
                   label = HTML("&alpha;<sub>12</sub>: 物种2对于物种1的人均影响"),
                   min = 0.0001, max = 1, value = 0.001, step = 0.0001),
      numericInput(inputId = "alpha22_manual",
                   label = HTML("&alpha;<sub>22</sub>:物种2对于自身的人均影响"),
                   min = 0.0001, max = 1, value = 0.005, step = 0.0001),
      numericInput(inputId = "alpha21_manual",
                   label = HTML("&alpha;<sub>21</sub>: 物种1对于物种2的人均影响"),
                   min = 0.0001, max = 1, value = 0.001, step = 0.0001)
    ),

    # User-defined initial values
    numericInput(inputId = "N1_wo_K",
                 label = "物种1的起始种群规模",
                 min = 1, value = 50),
    numericInput(inputId = "N2_wo_K",
                 label = "物种2的起始种群规模",
                 min = 1, value = 70),

    numericInput(inputId = "max_time_wo_K",
                 label = "模拟时长",
                 min = 1, value = 100)
  ),

  # Panel of plots 
  mainPanel(renderPlot(N_vs_Time_wo_K(), width = 600, height = 500),
            renderPlot(ZNGI_wo_K(), width = 600, height = 500))
)

# Store inputted competition coefficients as new reactive objects
alpha11_wo_K <- reactive({
  if (input$alpha_input_wo_K == "slider") {
    input$alpha11_slider
  } else if (input$alpha_input_wo_K == "manual") {
    input$alpha11_manual
  }
})
alpha12_wo_K <- reactive({
  if (input$alpha_input_wo_K == "slider") {
    input$alpha12_slider
  } else if (input$alpha_input_wo_K == "manual") {
    input$alpha12_manual
  }
})
alpha22_wo_K <- reactive({
  if (input$alpha_input_wo_K == "slider") {
    input$alpha22_slider
  } else if (input$alpha_input_wo_K == "manual") {
    input$alpha22_manual
  }
})
alpha21_wo_K <- reactive({
  if (input$alpha_input_wo_K == "slider") {
    input$alpha21_slider
  } else if (input$alpha_input_wo_K == "manual") {
    input$alpha21_manual
  }
})

# Get user-defined parameters
init_wo_K <- reactive({
  c(N1 = input$N1_wo_K, N2 = input$N2_wo_K)
})
time_wo_K <- reactive({
  seq(from = 0, to = input$max_time_wo_K, by = 0.1)
})
params_wo_K <- reactive({
  c(r1 = input$r1_wo_K, r2 = input$r2_wo_K,
    a11 = alpha11_wo_K(), a12 = alpha12_wo_K(),
    a22 = alpha22_wo_K(), a21 = alpha21_wo_K())
})



# Run lotka_volterra_competition_wo_K function
lvcomp_out_wo_K <- reactive({
  run_lvcomp_model(time = time_wo_K(), 
                   init = init_wo_K(), params = params_wo_K()) %>% 
    data.frame()
})

# Plot N vs time for both species
N_vs_Time_wo_K <- reactive({
  plot_lvcomp_time(lvcomp_out_wo_K()) 
})

# Plot ZNGIs with population trajectories
ZNGI_wo_K <- reactive({
  if ("ZNGI_wo_K" %in% input$plots_to_show_wo_K) {
    plot_lvcomp_portrait(lvcomp_out_wo_K(), params = params_wo_K())
  } 
})

# Make a list of plots to print based on user request
plot_list_wo_K <- reactive({
  list(N_vs_Time_wo_K(), ZNGI_wo_K()) %>%
    discard(is.null)
})

plots_to_print_wo_K <- reactive({
  wrap_plots(plot_list_wo_K(), ncol = 1)
})

参考文献

• Dr. Sarah Otto's lecture notes on Lotka-Volterra competition.

• Vandermeer and Goldberg, 2013. Population Ecology., Ch. 8. (Access to a digital copy of this textbook may be available through your library).

• Rosenzweig and MacArthur, 1963. Graphical Representation and Stability Conditions of Predator-Prey Interactions. The American Naturalist. Note: this paper focuses on predator-prey models rather than species competition, but it presents a clear explanation of the concept of zero net-growth isoclines.

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suppressWarnings(ecoevoapps::print_app_footer(language = "ch"))

gauravsk/ecoevoapps documentation built on July 9, 2024, 9:37 p.m.