Ecological systems in a modeling perspective

Abstract

May (1974,1976) opened the debate on whether biological populations might exhibit nonlinear dynamics and chaos. However, it has in general been difficult to verify nonlinear dynamics in biological populations. There are many reports concerning problems with this issue and some of them can be traced back to Hassell, Lawton, and May (1976) and Morris (1990). Our objective is not a discussion of the presence of nonlinear dynamics in biological populations. Instead, we analyze whether ecological census data can be used for validating nonlinearities at all. We choose our models and our situation so that as much as possible can be done rigorously with by hand computations. We consider a clearly nonlinear chemostat based model that is isolated. Some noise must be considered, and we choose a minimal approach: Only noise originating from the fact that ecological populations remain finite is considered, cf. Bailey (1964). In ecology, exceptionally long and famous time series are those collected by Nicholson (1954) and Utida (1957). Our judgement is that ecological time series data containing a few hundred data points is exceptionally long.

Figures (7)

• Figure 1: Experimental setup for a chemostat. It is assumed that the fluids in both containers are kept well-mixed. This basic setup can be extended to contain more containers, species, and nutrients.
• Figure 2: Sketch of the phase portrait of (\ref{['chemo_abundance']}). The parameters $M=3\cdot 10^{-22}$, $A=.01$, $B=0$, $I_0=8\cdot 10^{23}$, $D=.8$, and $L=1$ have been used. The expressions indicated for the different quantities in the figure are (if dependent on $B$) those corresponding to the case $B=0$. The slanted line is the asymptotically invariant line (\ref{['invariant-plane_deriv_prop']}).
• Figure 3: Quasi-stationary probability mass functions (red circles) compared to normal approximations sharing their expectations and variances (red dashed curves), to diffusion approximations (magenta dotted curves), and to moment closure approximations. The parameter values are $M=3\cdot 10^{-22}$, $A=.01$, $D=.8$ and we have (a) $I_0=I_0^\ast(0)\approx 2.923 \cdot10^{23}$, (b) $I_0=3\cdot10^{23}$, (c) $I_0=4\cdot10^{23}$, and (d) $I_0=6\cdot10^{23}$. The red lines correspond to the expectations, expectations plus/minus one standard deviation, and to expectations plus/minus two standard deviations. The magenta lines correspond to the expectations and standard deviations of the diffusion approximation. The black lines correspond to the expectations and standard deviations of the moment closure approximation.
• Figure 4: Phase portraits of (\ref{['Moment_close_ode']}). The parameter values are $M=3\cdot 10^{-22}$, $A=.01$, $D=.8$, $q_1=0$ and we have (a) $I_0=I_0^\ast(0)\approx 2.923 \cdot10^{23}$, (b) $I_0=3\cdot10^{23}$, (c) $I_0=4\cdot10^{23}$, and (d) $I_0=6\cdot10^{23}$. The blue and the red curves are the isoclines of (\ref{['Moment_close_ode']}) and the green curve is the stable manifold of the saddle point given by (\ref{['mom_closure_est']}). The saddle point and the stable fixed point collide and disappear at the saddle-node-bifurcation at $I_0=I_0^\ast(0)\approx 2.923 \cdot10^{23}$.
• Figure 5: Parameter values are given by $M=3\cdot 10^{-22}$, $A=.01$, $I_0=3\cdot 10^{23}$, $D=.8$, and $L=1$. (a) Quasi-stationary and stationary probability mass function of (\ref{['logistic_stok']}), (\ref{['linear_stok']}), and (\ref{['Gompertz_stok']}), marked with red, green, and blue $\circ$-marks, respectively, together with normal distributions sharing their mean and variance (dashed, same colors) and normal distributions derived as diffusion approximations (dotted, magenta for (\ref{['logistic_stok']}), (\ref{['linear_stok']})) and cyan blue for (\ref{['Gompertz_stok']}). (b) Simulation of (\ref{['logistic_stok']}) denoted with a yellow curve, census data denoted by red $\circ$-marks. (c) Census data from (\ref{['logistic_stok']}) denoted by red $\circ$-marks compared to the maps (\ref{['diskret_logistic']}), (\ref{['diskret_linear']}), and (\ref{['diskret_Gompertz']}) (d) AIC-values for a detectable nonlinearity in (\ref{['logistic_stok']}) (red dots), (\ref{['linear_stok']}) (green), and (\ref{['Gompertz_stok']}) (blue) versus the available amount of data.