Dynamics of a discrete-time stoichiometric optimal foraging model

Figure 1.  Attractor of discrete-time optimal foraging model (4) in phase plane for different light intensities in two cases. Panels of ($ a_i $) describe the case when the producer's growth rate is low $ b = 1.2 $, $ i = 1,2,3 $. Panels of ($ b_i $) describe the case when the producer's growth rate is high $ b = 3 $, $ i = 1,2,3 $. The red dashed curves are defined by $ F(x,y) = 0 $, which denote the producer nullclines. The blue dotted curves are defined by $ G(x,y) = 0 $, which denote the grazer nullclines. Solid bullets denote stable equilibria while circles represent unstable equilibria

Figure 2.  Bifurcation diagram of the population densities with respect to $ K $ (light intensity) for the discrete-time model (4) ($ a_i $), i = 1, 2, and the continuous-time model (1) ($ b_i $), i = 1, 2. Shaded regions with $ + $ represent the parameter regions of the optimal foraging behaviors benefiting the grazers. All parameters are provided in Table 1 with $ b = 1.2 $

Figure 3.  Solution curves for system (4) and (1). ($ a_i $) and ($ b_i $) denote the dynamics of (4) and (1) with increasing $ K $, respectively. Producer and grazer's densities ($ \mathrm{mg\; C/L} $) are plotted by dashed and solid lines, respectively. All parameters are provided in Table 1 with $ b = 1.2 $

Figure 4.  Solution curves for system (4) and (1). ($ a_i $) and ($ b_i $) denote the dynamics of (4) and (1) with increasing $ K $, respectively. Producer and grazer densities ($ \mathrm{mg\; C/L} $) are described by dashed and solid lines, respectively. All parameters are provided in Table 1 with $ b = 3 $

Figure 5.  The bifurcation curves with respect to $ K $ for the discrete-time model ($ a_i $), i = 1, 2, and continuous-time model ($ b_i $), i = 1, 2. All parameters are provided in Table 1 with $ b = 3 $

Figure 6.  Spectrum of the maximum Lyapunov exponent (MLE) with respect to $ K $ for the discrete-time model. All parameters are provided in Table 1 with $ b = 3 $

Figure 7.  A two-parameter bifurcation diagram for varying light level $ K $ and varying maximal growth rate of producer $ b $ for the discrete-time model (a) and continuous-time model (b). All other parameter values are listed in Table 1 and the initial point is $ x(0) = 0.2\; \mathrm{mg C L}^{-1} $ and $ y(0) = 0.2\; \mathrm{mg C L}^{-1} $. Discrete-time model (4) exhibits periodic oscillations in blue region and chaotic behaviors in red region. Outside these regions, model (4) has stable equilibria

Figure 8.  Bifurcation diagram of the grazer densities with respect to $ K $ (light intensity) for the discrete-time model (4) ($ a_i $), i = 1, 2, and the continuous-time model (1) ($ b_i $), i = 1, 2. Specially, ($ a_1 $) and ($ b_1 $) denote the case with the low growth rate of producer ($ b = 1.2 $); ($ a_2 $) and ($ b_2 $) denote the case with the high growth rate of producer ($ b = 3 $). Light ($ + $) and dark ($ - $) shaded regions represent the parameter regions of the optimal foraging behaviors benefiting and restraining the grazers, respectively. All parameters are provided in Table 1 except the parameter $ a_1 = 3.5 $