Zooplankton selectivity and nutritional value of phytoplankton influences a rich variety of dynamics in a plankton population model

Mathematical modeling may be an excellent tool to analyze and explain complex biological phenomena. In this paper, we use a mathematical model to reveal various interesting dynamical features of phytoplankton-zooplankton interaction and attempt to explain the reason for contrasting dynamics shown by different laboratory and field experiments. Our study shows that the phytoplankton-zooplankton interaction in a pelagic system is very complex and the plankton dynamics, including the bloom phenomenon, strongly depends on the selective predation of zooplankton and the nutritional value of phytoplankton. The study supports the existing hypothesis that decoupling at the plant-animal interface may occur due to strong fish predation on zooplankton. In addition, we argue that decoupling of the food chain may also occur under low to intermediate nutrient inflow if zooplankton feeds on phytoplankton having lower nutritional value. It is also shown that nutrient enrichment can destabilize an otherwise stable system if zooplankton feeds on highly nutritious prey, but unable to destabilize the system if zooplankton feeds on low-nutritious prey. This may be one possible explanation to the longstanding question: Why do some experiments show the paradox of enrichment and others do not?

Figure 1

Bifurcations of plankton dynamics against preference parameter $\theta$ in panels (a)–(c). Five distinct varieties of dynamical behaviors are seen as $\theta$ varies: (1) three populations $p_1, p_2$, and $z$ coexist in oscillatory states (blue line) for $\theta >1.712$, (2) $p_1$ and $z$ coexist in oscillatory states [green lines in panels (a) and (c)] but $p_2$ goes to extinction [green line in panel (b)] in the range $1.1077<\theta <1.712$, (3) NP-free equilibrium $E_4$ is stable (red line) for $\theta \in (0.4144,1.1077)$ where $p_1$ and $z$ have constant densities but $p_2$ has zero density, (4) three species coexist at steady states in a smaller range $0.4144<\theta <0.3975$, and (5) zooplankton-free equilibrium $E_3$ is stable for $0<\theta <0.4144$, where $p_1$ and $p_2$ have constant densities and $z$ has zero density. Red dotted lines represent stable fixed points, green and blue lines for the limit cycle and, black dots represent unstable fixed points. A transcritical bifurcation of the limit cycle is depicted by the transition from the blue to green line at $\theta =1.712$. The junctions of green and red lines indicate the Hopf bifurcation point at $\theta =1.1077$. A first transcritical bifurcation of fixed point occurs at $\theta =0.4144$, where the NP population $p_2$ revives from extinction and all species coexist in steady state for a shorter range. At $\theta =0.3975$, a second transcritical bifurcation of fixed point occurs when $z$ goes to extinction and NP $\left(p_2\right)$ maintains a low density along with high density of PP $\left(p_1\right)$. (d) The 3D trajectory of all species more clearly depicts the transcritical bifurcation of a limit cycle. The trajectory in the green line lies on $p_2=0$ plane, but the trajectory in blue shows the coexistence of all three species, indicating an exchange of stability of the limit cycles. Parameters are as in Table 1 except $\theta$. No fish predation is considered, i.e., $D=0$.

Figure 2

Variety of temporal dynamics of plankton population in system (1) for particular values of preference parameter $\theta$. The populations of $p_1, p_2$, and $z$ are shown in black, red, and blue lines, respectively. As $\theta$ varies from higher to lower values, the system exhibits (a) relaxation oscillation around $E^{*}$ when all three populations oscillate for $\theta >1.712$, (b) regular oscillation around $E_4$ where PP and zooplankton oscillate but NP goes to extinction for $1.712\ge \theta >1.1077$, (c) $E_4$ is stable where PP and zooplankton have constant density but NP goes to extinction, for $0.4144<\theta <1.1077$, (d) three species coexist in a stable steady state $E^{*}$ for $0.3975<\theta <0.4144$, and (e) $E_3$ is stable where PP and NP have constant density yet zooplankton goes to extinction, for $0<\theta <0.3975$. Parameters are considered as given in Fig. 1.

Figure 3

Biomass change with nutritional value of NP when zooplankton feeds on PP $(\theta =2)$ is presented in left panels. Right panels show the biomass change with nutritional value of NP when zooplankton feeds on NP $(\theta =0.5)$. Green lines indicate maxima and minima of oscillatory dynamics, red lines for stable steady states, and black lines for unstable steady states. Parameters are as in Table 1 except ${\xi }_2$. No fish predation is considered ($D=0$).

Figure 4

Plankton dynamics under two different preferences when the removal rate of zooplankton augmented by fish predation is varied. Figures in the left panel show the dynamics when zooplankton is inclined to PP $(\theta =2)$ and in the right panel represent the same when zooplankton is inclined to NP $(\theta =0.5)$. Green line indicates oscillatory state; red and black dots indicate stable and unstable steady states, respectively. A reverse Hopf bifurcation occurs at the junction of the green line and red dots at $d_1=0.1574$, and a transcritical bifurcation occurs at $d_1=0.1861$ [see panels (a)–(c)]. Figures in the right panel show two consecutive transcritical bifurcations at $d_1=0.1293$ (transition from $E_4$ to $E^{*}$) and $d_1=0.1391$ (transition from $E^{*}$ to $E_3$). Parameters are as in Table 1 except $d_1$.

Figure 5

Bifurcation of plankton ($p_1, p_2$, and $z$) dynamics with nutrient variation under two preferences of zooplankton: $\theta =2$ (left panels) and $\theta =0.5$ (right panels). Green lines indicate a stable oscillatory state; red and black dots denote stable and unstable steady states, respectively. Recurrence multispecies bloom are the most probable outcomes of the system when zooplankton prefers PP $(\theta =2)$ with increasing nutrient availability. If zooplankton prefers NP $(\theta =0.5),$ then zooplankton-free stable state or NP-free stable state are the most likely cases as nutrient inflow is increased. Parameters are as in Table 1 except $k_1=k_2=k$.

Figure 6

Two parameters bifurcation on the $\theta -k$ plane. Equilibrium points $E_3, E_4$, and $E^{*}$ are stable in the $R_3, R_4$, and $R^{*}$ regions, respectively. $E_4$ and $E^{*}$ become unstable through Hopf bifurcation and exhibit oscillations in the regions $R_4^{{}^{\prime }}$ and $R^{{*}^{\prime }}$, respectively.

Figure 7

Two parameters bifurcation in $\theta -d_1$ plane. Orange, yellow, and red regions denote stable regions of $E_3, E_4$, and $E^{*}$, respectively. The oscillatory state is denoted by the white region.

Figure 8

Two parameters bifurcation in the $k-d_1$ plane for (a) $\theta =0.5$ and (b) $\theta =2$. (a) The orange region is for stable equilibrium $E_4$, red region for $E^{*}$, and yellow for $E_3$. (b) Dark yellow $E_3$, red $E_4$, orange $E^{*}$, yellow $E_1$. The white region is for limit cycle oscillations.