Discrete population logistic growth model. $$ x_{t+1}=r x_t(1-x_t) $$ $x$ stands for population density.
Running examples
$r=2,3,3.5,3.7, 3.8, 4$
Biblography: Murray, J.D., Mathematica biology. I. An Introduction, 3rd ed., Springer, 2002.
Predator-Pray interaction model with non-ovelaping generations. $$ \begin{array}{l} x_{t+1}=x_t \exp{(r(1-x_t)-a y_t)}\\ y_{t+1}=x_t (1-\exp{(-a y_t)}) \end{array} $$ where $x$ stands for the pray density and $y$ stands for predator density.
Running examples:
$(r,a)=(3,3.1),(3,3.3), (3,3.6), (3,3.7), (3,4), (3,4.2)$
Initial state: $(x0,y0)=(0.1,0.2)$
Biblography: Murray, J.D., Mathematica biology. I. An Introduction, 3rd ed., Springer, 2002.
Vegetation Model is a mathematical ODE model that tray to explain how the water influence the vegetation growth, here is a model without diffusion terms: $$ \begin{array} \displaystyle\frac{d x}{d t}=cg_m\displaystyle\frac{ y}{y+k_1}x-d_1x\\ \displaystyle\frac{d y}{d t}=\alpha\displaystyle\frac{x+k_2 y_0}{x+k_2}z-g_m\displaystyle\frac{ y}{y+k_1}x-i_wy\\ \displaystyle\frac{d z}{d t}=r_w-\alpha\displaystyle\frac{x+k_2 y_0}{x+k_2}z\ \end{array} $$ where $x$ denotes the plant density, $y$ stands for soil water and $z$ denotes the surface water.
The parameters in the model are:
$c$ - the factor conversion of water uptake by plant to plant growth,
$g_{m}$ - maximum water uptake,
$d_1$ - rate of plant mortality,
$\alpha$ - maximum water infiltration rate,
$i_w$ - rate of soil water loss,
$y_0$ - rate of water infiltration in the absence of the plant,
$r_w$ - the rainfall,
$k_1$ and $k_2$ the modelling constants.
Equilibrium points:
$$\left(0,\frac{r_w}{i_w}, \frac{r_w}{\alpha y_0}\right); \left(x,y, r_w\frac{x+k_2}{\alpha (x+k_2 y_0)}\right); x=\frac{c}{d_1}(r_w-i_w y);y=\frac{k_1d_1}{cg_m-d_1}$$ On graphics: $x$ on vertical axis and $y$ on horizontal axis.
Fixed parameters: $(c,g_m,d_1,\alpha,y_0,k_1,k_2)=(10,0.05,0.25,0.2,0.2,5,5);
Initial state is set at (0.1,1,1).
Runnig examples:
Free Parameters: $(r_w,i_w)=(1.3,0.24);(1.3,0.2575)$;
Klausmeir, C.A, Regular and irregula patterns in semiarid vegetation, Science, 284, pp. 1826-1828
Max Rietkerk et al., Self-Organization of vegetation in Arid Ecosystems, The American Naturalist, 160(4), 2002, pp. 524-530.
Predator - pray model with strong Allee efect, Owen-Lewis Model with zero diffusion:
$$ \begin{array} \displaystyle\frac{d x}{d t}=x(1-x)(\displaystyle\frac{x}{b}-1)-\displaystyle\frac{mxy}{a+x},0\leq b<\leq 1\\ \displaystyle\frac{d y}{d t}=-d*y+\displaystyle\frac{mxy}{a+x}\\ \end{array} $$
$x$ stands for pray density, $y$ stands for predator density.
Equilibrium points:
$$(0,0),(b,0),(\beta,\displaystyle\frac{(a+\beta)(1-\beta)(\beta-b)}{mb})$$ where $\beta=\displaystyle\frac{ad}{m-d} and d\leq m, b\leq\beta\leq 1 $
Runnig examples:
The parameter $a$ is set $a=b+1$.
Parameters: $b=0.5$, m=1,d=0.336$,
Initial state (0.2,0.1), (0.7,0.1)
P.A. Stephen, W.J. Sutherland, R.P. Freckleton, What is the Allee effect ?, OIKOS 87:1 (1999)
M.R. Owen, M.A. Lewis, How Predation can Slow, Stop or Reverse a Pray Invasion, Bull. Math. Biology, 63(2001).