Interaction between water and plants: Rich dynamics in a simple model

Figure 1.  A water-plant interaction system with the infiltration feedback

Figure 2.  The graph of biomass per capita death rate $\mu(b)=\mu_0+\frac{\mu_1}{b+1}$. Here $\mu_0=0.5$, $\mu_1=2$, and $0\le b\le 10$

Figure 3.  Possible bifurcation diagrams for (1) - (2) for different $(\lambda ,{\mu _0},{\mu _1})$. In all diagrams, the horizontal axis is the rain-fall rate $R$, and the vertical axis is the biomass $b$. (a) forward bifurcation and no cycle; (b) forward bifurcation and bubble loop of cycles; (c) backward bifurcation and no cycle; (d) backward bifurcation and bubble loop of cycles; (e) backward bifurcation, and bubble loop of cycles with one subcritical and one supercritical Hopf bifurcations and a saddle-node bifurcation of cycles; (f) backward bifurcation, subcritical Hopf bifurcation, and branch of cycles into a homoclinic bifurcation; (g) backward bifurcation, supercritical Hopf bifurcation, and branch of cycles into a homoclinic bifurcation; (h) backward bifurcation, subcritical Hopf bifurcation, saddle-node bifurcation of cycles and branch of cycles into a homoclinic bifurcation

Figure 4.  Illustration of the stability parameter subregions in the $\mu_0-\mu_1$ plane. (Left) $0< \lambda<1$; (Right) $\lambda\ge 1$

Figure 5.  Bifurcation diagrams (bubble branch of cycles) and phase portraits of limit cycles when there are two Hopf bifurcation points $R=R_2$ and $R=R_3$. In (a), (c) and (e), the blue curve, the cyan curve and the purple curve represent the biomass $b$, the minimum value and the maximum value of $b$ of the limit cycles versus the rain-fall rate $R$, respectively; corresponding phase portraits are shown in (b), (d) and (f). In (a) and (b) $\lambda=0.2$, $(\mu_0, \mu_1)=(7, 4.7)\in \textrm{Ⅲ}$, and $R=11.14$ in (b); in (c) and (d) $\lambda=0.2$, $(\mu_0, \mu_1)=(20, 4.8)\in \textrm{V}$, and $R=26$ in (d); and in (e) and (f) $\lambda=1.2$, $(\mu_0, \mu_1)=(5, 8)\in \textrm{Ⅶ}$, and $R=16$ in (f)

Figure 6.  Hopf bifurcation and multiple limit cycles when $\lambda=0.2$ and $(\mu_0, \mu_1)\in \textrm{Ⅲ}$. (a): The bifurcation diagram when $(\mu_0, \mu_1)=(20, 6.5)\in \textrm{Ⅲ}$. The blue curve, the cyan curve and the purple curve represent the biomass $b$, the minimum value and the maximum value of $b$ of the limit cycles vs. the rain-fall rate $R$, respectively. (b): Two limit cycles when $(\mu_0, \mu_1)=(20, 6.5)\in \textrm{Ⅲ}$ and $R=31.4$. (c): Period of limit cycles versus $R$ when $(\mu_0, \mu_1)=(7, 4.7)\in \textrm{Ⅲ}$, and the period is monotonically decreasing in $R$. (d): Period of limit cycles versus $R$ when $(\mu_0, \mu_1)=(20, 6.5)\in \textrm{Ⅲ}$, and the period is not monotone in $R$ and is not single-valued (indicating multiple limit cycles). (e): Time series of the small amplitude periodic orbit when $(\mu_0, \mu_1)=(20, 6.5)\in \textrm{Ⅲ}$ and $R=31.4$. (f): Time series of the large amplitude periodic orbit when $(\mu_0, \mu_1)=(20, 6.5)\in \textrm{Ⅲ}$ and $R=31.4$

Figure 7.  Homoclinic bifurcation, saddle-node bifurcation, Hopf bifurcation and the existence of one limit cycle when $\lambda=0.2$ and $(\mu_0,\mu_1)\in \textrm{Ⅱ}$. In (a), (c), the blue curve, the cyan curve and the purple curve represent the biomass $b$, the minimum value and the maximum value of $b$ of the limit cycles vs. the rain-fall rate $R$, respectively. (a): Bifurcation diagram when $(\mu_0,\mu_1)=(5,5.2)\in \textrm{Ⅱ}$. (b): Phase portrait with periodic orbit when $(\mu_0,\mu_1)=(5,5.2)$ and $R=9.17$. (c): Bifurcation diagram when $(\mu_0,\mu_1)=(7,4.9)\in \textrm{Ⅱ}$. (d): Phase portrait with periodic orbit when $(\mu_0, \mu_1)=(7,4.9)$ and $R=11.38$. (e): Period of the stable periodic orbits when $(\mu_0,\mu_1)=(5,5.2)$. (f): Period of the unstable limit cycles when $(\mu_0,\mu_1)=(7,4.9)$

Figure 8.  Evolution of phase portraits of (3) for $\lambda=0.2$, $(\mu_0, \mu_1)=(5,5.2)\in \textrm{Ⅱ}$ and $R>R_1$. (a) $R=9.14$; (b) $R=9.17$; (c) $R=9.1830408$; (d) $R=9.19$

Figure 9.  Evolution of phase portraits of (3) for $\lambda=0.2$ and $(\mu_0, \mu_1)=(7,4.9)\in \textrm{Ⅱ}$ and $R>R_1$. (a) $R=11.3$; (b) $R=11.363757$; (c) $R=11.38$; (d) $R=11.4$

Figure 10.  Homoclinic bifurcation, saddle-node bifurcation, Hopf bifurcation and multiple limit cycles for $\lambda=0.2$ and $(\mu_0,\mu_1)=(7,5)\in \textrm{Ⅱ}$. (a): The bifurcation diagram. The blue curve, the cyan curve and the purple curve represent the biomass $b$ bifurcation diagram, the minimum value and the maximum value of $b$ of the limit cycles vs. the rain-fall rate $R$, respectively. (b): Phase portrait with one stable limit cycle when $R=11.48$. (c): Period of limit cycles versus $R$. (d): Phase portrait with two limit cycles when $R=11.498$. (e): Time series of the small amplitude periodic orbit when $R=11.498$. (f): Time series of the large amplitude periodic orbit when $R=11.498$

Figure 11.  Evolution of phase portraits of (3) for $\lambda=0.2$, $(\mu_0, \mu_1)=(7, 5)\in \textrm{Ⅱ}$ and $R>R_1$. (a) $R=11.4$; (b) $R=11.47377$; (c) $R=11.48$; (d) $R=11.498$; (e) $R=11.5$; (f) $R=12$

Figure 12.  The cyan curve represents the Hopf bifurcation curve, the blue curve represents the saddle-node bifurcation curve and the black line is $\mu_1=R-\mu_0$. The "BT" mark indicates a Bogdanov-Takens bifurcation point; the "CP" mark indicates a cusp bifurcation point; and the "GH" mark indicates a generalized Hopf point where the first Lyapunov coefficient vanishes while the second Lyapunov coefficient does not vanish, which indicates that it is nondegenerate, i.e. Hopf bifurcation changes from subcritical to supercritical. (a): $\lambda=0.2, \mu_0=5$. (b): $\lambda=0.2, \mu_0=7$. (c): $\lambda=0.2, \mu_0=20$. (d): $\lambda=1.2, \mu_0=5$