Seasonal refuge patterns of phytoplankton trigger irregular bloom events in a contaminated environment

Abstract

Several experimental evidences and field data documented that zooplankton may alter its behavioral response in the presence of toxic phytoplankton, reducing its consumption to the point of starvation. This paper is devoted to the mathematical study of such interactions of toxic phytoplankton with grazer zooplankton. The non-toxic phytoplankton is assumed to adopt a density-dependent refuge strategy to avoid over-predation by zooplankton. Both groups of phytoplankton are assumed to suffer direct harm from anthropogenic toxicants, while zooplankton is affected indirectly by ingesting contaminated phytoplankton. We calibrate the proposed model with the field data from Talsari and Digha Mohana, India, and estimate some crucial model parameters consistent with the behavior of the observed data. Our results demonstrate that zooplankton grazing on toxic phytoplankton plays a key role in the emergence or mitigation of plankton blooms. We also highlight the system’s potential to exhibit multiple stable configurations under the same ecological conditions. The plankton system experiences significant regime shifts, which are explored through various bifurcation scenarios, such as transcritical and saddle-node bifurcations. These shifts are influenced by changes in refuge capacity, species growth rates, and environmental carrying capacity. Furthermore, we incorporate environmental variations due to seasonal periodic or almost periodic changes, allowing the refuge parameter to be time-dependent. We observe that the forced system exhibits double periodic solutions. Moreover, stronger seasonal variations in the refuge pattern lead to irregular chaotic blooms. In conclusion, the results offer valuable insights into the sustainability of biodiversity, potentially shedding light on the origin of diverse plankton bloom phenomena.

Introduction

The role of phytoplankton in ocean ecosystems is of critical importance due to their participation in the extensive global processes such as climate regulation, global carbon cycle, and ocean-atmosphere dynamics. Phytoplankton forms the base of the food chain in lakes, oceans, reservoirs, etc., and is identified as the energy source of the aquatic food webs. The presence of a favorable environment, including factors like water temperature, salinity levels, and inorganic nutrients, plays a key role in promoting the rapid growth of phytoplankton biomass. The development of mathematical models to investigate the dynamics of bloom events and their underlying mechanisms is of utmost significance and dominates the field of ecology^1,2.

Certain groups of phytoplankton have the ability to release “allelopathic” or “toxic substances” (e.g., Noctiluca scintillans, Alexandrium catenella, Dinophysis acuminata, etc.), which can be transferred through the food web and impact marine and terrestrial species³. Turner and Tester⁴ highlighted the complex relationship between toxic phytoplankton and zooplankton grazing. They noted that these interactions can have contrasting outcomes. In some instances, zooplankton graze on toxic phytoplankton without any harm. However, in other cases, consumption can trigger severe damage. Studies have shown that the copepods Acartia bifilosa and Eurytemora affinis can exhibit very slow feeding rates on the cyanobacterium Nodularia spumigena, leading to starvation in some cases. Acartia bifilosa appears to be particularly susceptible, experiencing high mortality rates⁵. The release of toxic compounds impact the feeding patterns and morphology of zooplankton, ultimately causing a considerable decline in their biomass over time. Such an anti-predator defense helps to curtail the chaotic disorder of planktonic species and shapes the devastating bloom dynamics⁶. Several modeling frameworks have been used to provide quantitative insights into the dynamics of seasonally mediated bloom phenomena^2,7. Chattopadhyay et al.⁸ established that toxic chemicals liberated by phytoplankton possess a pivotal role in the mitigation of algal bloom.

The discharge of industrial waste, agricultural and urban expansion, accidental spillage, and related factors can pose potential threats to aquatic life and ecosystems. Exposure to toxins can unfavorably alter the physiological characteristics of algae cells, potentially impeding their growth, and in severe cases, causing acute mortality. Suspended industrial waste and other potential pollutants disrupt the Hill reaction of algae photosynthesis by obstructing the penetration of light into the water column⁹. As a result, many of these contaminants can significantly reduce the uptake rates of ammonia and nitrate by phytoplankton. These toxic substances accumulate in higher trophic levels of the food chain through predation. Research by Sprules¹⁰ indicated that industrial acidification brings about substantial shifts in the composition of crustacean zooplankton species due to a pH decrease from 7.0 to 3.8. The mathematical exploration of the indirect effects of toxicants on zooplankton through the consumption of contaminated phytoplankton has been documented in^11,12. In a contaminated medium, Biswas et al.¹³ examined the selective feeding behavior of zooplankton on phytoplankton with viral load. Their findings indicated that a high abundance of environmental contaminants leads to the extinction of the zooplankton population. Additionally, Mandal et al.¹⁴ examined the influence of public awareness on anthropogenic effluents and their impact on the planktonic system.

The capability to differentiate between non-toxic and toxic resources can lower the likelihood of zooplankton extinction. Zooplankton demonstrates chemotactic sensitivity by moving against the concentration gradient of dense toxic phytoplankton (TPP), thereby reducing its intake to the point of potential starvation. This phenomenon is exemplified by the “Exclusion Principle”. Gragnani et al.¹⁵ documented that the discerning predatory behavior of zooplankton, driven by the nutritional differences among planktonic species and the intricate food web dynamics within marine ecosystems, critically contributes to extending the coexistence and maintaining the biodiversity of phytoplankton and zooplankton populations.

In response to the evading risk of predation, prey species actively embrace a defense against the predators by taking refuge. Greater predation risk often results in a stronger shift of prey into less-accessible zones in the form of refuge^16,17,18. The use of spatial refuges by the prey is one of the most fascinating behavioral traits that shape the community structure and regulate the dynamics of predator-prey interactions. This hiding behavior provide themselves some degree of protection from potential predators and reasonably promotes the chance of prey survivability. Phytoplankton can be dispersed over large areas due to water currents. This form of passive dispersal can aid in minimizing the impact of localized predation. Schindler et al.¹⁹ showed that benthic sediments can potentially provide the phytoplankton species with a refuge for egg production or be dormant, contributing to a temporal escape from zooplankton predation. According to Wiles et al.²⁰, water column stratification can serve as a temporary sanctuary for phytoplankton species, allowing them to recuperate from the pressures of predation. Refugia may exert a stabilizing influence on the system dynamics by diminishing the oscillatory behaviors of prey and predator populations²¹. In contrast, McNair²² documented that refuge may possess a destabilizing role by producing periodic solutions. Spatial refuge enhances the persistence of the system and decreases the probability of prey extinction. Haque and Sarwardi²³ constructed a mathematical model for predator-prey system by taking into account the dependence of functionality of prey refuge on both the species. In a Recent study, Mandal et al.²⁴ documented that this type of density-dependent refuge used by zooplankton in a plankton-fish system can lead to multistability. Li et al.²⁵ investigated a simple phytoplankton-zooplankton system by considering that phytoplankton take refuge and also release toxins, and showed that both refuge and toxin may act as bio-control units for the onset and termination of noxious blooms. Nonetheless, the efficacy of plankton refuge is contingent on diverse environmental factors like water temperature, light intensity, wind, rainfall, etc., which undergo seasonal variations in a cyclic pattern. This suggests that incorporating time-dependent refuge by phytoplankton into modeling is a pragmatic approach for comprehensively investigating the dynamics of a planktonic system within a fluctuating environment^7,11,21.

Several studies have been conducted to explore the interactions among non-toxic phytoplankton, toxic phytoplankton and zooplankton in presence of toxin allelopathy^1,26,27. Previously, Li et al.²⁵ studied the effect of a constant refuge capacity of phytoplankton in phytoplankton-zooplankton system. Mandal et al.¹¹ investigated the effect of refuge utilization by a constant fraction of the phytoplankton population against zooplankton. Within real plankton ecosystems, toxic phytoplankton can employ their toxicity as a defense mechanism, while their non-toxic counterparts find protection through a refuge mechanism. The extent of refuge is influenced by both phytoplankton and zooplankton populations. However, the impact of density-dependent phytoplankton refuge on the overall plankton ecosystem remains to be explored. As far as our understanding extends, there is a scarcity of models that consider the interaction between phytoplankton, toxic phytoplankton, and zooplankton in a contaminated environment, encompassing these distinct defense mechanisms. Motivated by this, we construct a mathematical framework that encompasses both toxic and non-toxic phytoplankton, along with grazer zooplankton, all subjected to varying levels of environmental toxins. We postulate that the non-toxic species utilize a temporary refuge mechanism to shield themselves from grazer zooplankton. Conversely, the toxic counterpart excretes allelopathic agents that exert a repellent effect on zooplankton, leading to the avoidance of dense populations of the toxic species.

Our study aims to address the following inquiries:

• 1. What is the impact of nutrient enrichment and the density-dependent mechanism of NTP refuge on the bifurcation characteristics and stability patterns of a plankton system exposed to contamination?

• 2. How do seasonal variations in the refuge pattern of NTP regulate the dynamics of devastating blooms?

In this context, we aim to address these inquiries by showcasing comprehensive scenarios that elucidate the intricate dynamics of phytoplankton-zooplankton interactions utilizing established tools from the field of nonlinear dynamics.

The mathematical model

Figure 1

Schematic diagram for the interactions among NTP, TPP and zooplankton. Here, the head on arrows of the dashed lines show the loss in both TPP and zooplankton biomasses due to consumption and assimilation of the former by the latter, the red color represents the effect of seasonality in the refuge pattern of NTP.

We present a mathematical framework for an aquatic ecosystem comprising two groups of phytoplankton, non-toxic and toxic, along with their grazer zooplankton. Let Z(t), $P_1(t)$, and $P_2(t)$ respectively denote the zooplankton, non-toxic phytoplankton and toxic phytoplankton groups at any time $t>0$. Our model is formulated based on the following biologically reasonable assumptions.

• 1. Both the toxic phytoplankton (TPP) and non-toxic phytoplankton (NTP) follow logistic growth in the absence of the other; when both appear, they actively compete for the available resources.

• 2. The NTP is capable of taking refuge depending on both NTP and zooplankton species, i.e., mPZ amount of NTP is free from predation²³, where m is the coefficient of refuge. Incorporation of prey refuge leaves $(1-mZ)P$ unprotected NTP available for grazer zooplankton following Holling type II functional response, i.e., $\displaystyle f_1(P_1)=\frac{\beta _1(1-mZ)P_1}{a_1+(1-mZ)P_1}$, where $\beta _1$ is the NTP uptake rate by zooplankton and $a_1$ is the half-saturation constant. For a realistic plankton system, $(1-mZ) \ge 0$, i.e., $Z \le 1/m$, which ensures the permissible range $0 \le (1-mZ)\le 1$ of NTP refuge.

• 3. Zooplankton recognizes the two phytoplankton populations and overconsumption of toxic species affects zooplankton to a great extent. Zooplankton exhibits a group defense by reducing TPP uptake, which is captured by Monod-Haldane functional form, i.e., $\displaystyle f_2(P_2)=\frac{\lambda _2 \beta _2P_2}{a_2+P^2_2}$²⁷.

• 4. Phytoplankton and zooplankton experience varying levels of pollutant exposure. We posit that the toxic substances directly affect both phytoplankton species, while zooplankton is impacted indirectly through the consumption of contaminated phytoplankton¹². Let $u_1$, $u_2$ and v $(0<v<u_1,u_2<1)$ be the coefficients of toxicity efficiency on the non-toxic, toxic phytoplankton and zooplankton populations, respectively. The term $u_{i}P_{i}^3$ measures reduction of phytoplankton populations by some external toxic substances. Since $\displaystyle \frac{d(u_{i}P^3_{i})}{dt}=3u_{i}P_{i}^2>0$ and $\displaystyle \frac{d^2(u_{i}P_{i}^3)}{dt^2}=6u_{i}P_{i} >0$, therefore there is an accelerated growth in the production of the toxic substances relative to the densities of phytoplankton populations, as both the species increasingly consume the contaminated foods. This term of toxicity was first considered by Das et al.²⁸. To calibrate the comparatively lesser impacts of toxicity on zooplankton over phytoplankton, we take $vZ^2$ as the reduction term for the zooplankton species¹².

Given these assumptions, a schematic diagram is illustrated in Fig. 1, and the corresponding model equations are as follows:

$$\begin{aligned} \frac{dP_1}{dt}= & {} r_1P_1\left( 1-\frac{P_1+\alpha _1 P_2}{K}\right) -\frac{\beta _1(1-mZ)P_1Z}{a_1+(1-mZ)P_1}-u_1P^3_1,\nonumber \\ \frac{dP_2}{dt}= & {} r_2P_2\left( 1-\frac{P_2+\alpha _2 P_1}{K}\right) -\frac{\beta _2P_2Z}{a_2+P^2_2}-u_2P^3_2,\nonumber \\ \frac{dZ}{dt}= & {} \frac{\lambda _1 \beta _1(1-mZ)P_1Z}{a_1+(1-mZ)P_1}-\frac{\lambda _2 \beta _2P_2Z}{a_2+P^2_2}-\mu Z-vZ^2. \end{aligned}$$

(1)

The analysis of system (1) will be conducted using the following initial conditions:

$$\begin{aligned} P_1(0)>0, \ P_2(0)>0, \ Z(0)>0. \end{aligned}$$

(2)

Mathematical analysis

0.0.1 Positivity and boundedness of solutions

Proposition 1

Every solution of model (1), subject to non-negative initial conditions (2), remains uniformly bounded within the subsequent region:

$$\begin{aligned} \Omega =\left\{ (P_1,P_2,Z)\in {\mathbb {R}}^3: 0 \le P_1(t),P_2(t) \le K, \ 0\le P_1(t)+P_2(t)+Z(t)\le N\right\} , \end{aligned}$$

which is invariant and compact with respect to system (1).

Proof

   Any solution $(P_1(t),P_2(t),Z(t))$ of system (1) with initial condition $(P_1(0),P_2(0),Z(0))$ satisfies,

$$\begin{aligned}{} & {} P_1(t)=P_1(0)\exp \left\{ \int ^t_0\left[ r_1\left( 1-\frac{P_1(s)+\alpha _1 P_2(s)}{K}\right) -\frac{\beta _1(1-mZ(s))Z(s)}{a_1+(1-mZ(s))P_1(s)}-u_1P^2_1(s)\right] ds\right\} ,\\{} & {} P_2(t)=P_2(0)\exp \left\{ \int ^t_0\left[ r_2\left( 1-\frac{P_2(s)+\alpha _2 P_1(s)}{K}\right) -\frac{\beta _2Z(s)}{a_2+P^2_2(s)}-u_2P^2_2(s)\right] ds\right\} ,\\{} & {} Z(t)=Z(0)\exp \left\{ \int ^t_0\left[ \frac{\lambda _1 \beta _1(1-mZ(s))P_1(s)}{a_1+(1-mZ(s))P_1(s)}-\frac{\lambda _2 \beta _2P_2(s)}{a_2+P^2_2(s)} -vZ(s)-\mu \right] ds\right\} . \end{aligned}$$

In view of the above expressions, the conclusion readily follows for all $t\in [0,+\infty )$.

In view of the positivity of the variables, from the first two equations of system (1), we get

$$\begin{aligned} \frac{dP_1}{dt}\le r_1P_1\left( 1-\frac{P_1}{K}\right) , \ \frac{dP_2}{dt} \le r_2P_2\left( 1-\frac{P_2}{K}\right) . \end{aligned}$$

(3)

Using the results for differential inequalities²⁹, it follows that

$$\begin{aligned} \limsup _{t\rightarrow \infty }P_1(t)\le K, \ \limsup _{t\rightarrow \infty }P_2(t)\le K. \end{aligned}$$

To show the boundedness of Z(t), we define a new function $U(t)=P_1(t)+P_2(t)+Z(t)$. For an arbitrary $\sigma >0$, calculating the time derivative of U(t), we obtain

$$\begin{aligned} \frac{dU}{dt}+\sigma U \le (r_1+\sigma )P_1+(r_2+\sigma )P_2-\frac{r_1P^2_1}{K}-\frac{r_2P^2_2}{K}-(\mu -\sigma )Z. \end{aligned}$$

For $\sigma \le \mu$, from the above inequality, we get

$$\begin{aligned} \frac{dU}{dt}+\sigma U \le (r_1+\sigma )P_1+(r_2+\sigma )P_2-\frac{r_1P^2_1}{K}-\frac{r_2P^2_2}{K}. \end{aligned}$$

Hence, we obtain

$$\begin{aligned} \frac{dU}{dt}+\sigma U \le \frac{K(r_1+\sigma )^2}{4r_1}+\frac{K(r_2+\sigma )^2}{4r_2}=M \ \text {(say)}. \end{aligned}$$

Applying standard results on differential inequalities²⁹, we get

$$\begin{aligned} U(t)\le e^{\displaystyle -\sigma t}\left( U(0)-\frac{M}{\sigma }\right) +\frac{M}{\sigma }. \end{aligned}$$

Thus, as $t\rightarrow \infty$, we have for some N

$$\begin{aligned} 0 \le U(t) \le \frac{M}{\sigma }=N. \end{aligned}$$

Therefore, the interacting species involved in system (1) are bounded above. $\square$

Given the nonlinearity inherent in the plankton system (1), exact solutions for the system are nearly unattainable. Instead, we focus on investigating the long-term dynamic trends within the system. Generally, a system of nonlinear equations either gravitates towards a steady state or it blows up.

System’s equilibria and stability

Now, we study the possible ecologically feasible steady states of the plankton system and investigate their local stability behaviors.

System (1) has the following six non-negative ecologically meaningful equilibria.

• 1. The plankton-free equilibrium $E_0=(0,0,0)$, which always exists.

• 2. The TPP-zooplankton-free equilibrium $E_1=({\widehat{P}}_1,0,0)$, where

  $\displaystyle {\widehat{P}}_1=\frac{-r_1+\sqrt{r_1(r_1+4K^2u_1)}}{2Ku_1}.$ The equilibrium $E_1$ always exists.

• 3. The NTP-zooplankton-free equilibrium $E_2=(0,{\widetilde{P}}_2,0)$, where

  $\displaystyle {\widetilde{P}}_2=\frac{-r_2 +\sqrt{r_2(r_2+4K^2u_2)}}{2Ku_2}.$ The equilibrium $E_2$ always exists.

• 4. The zooplankton-free equilibrium $E_3=({\overline{P}}_1,{\overline{P}}_2,0)$, where ${\overline{P}}_2$ is given by

  $$\begin{aligned} {\overline{P}}_2=\frac{1}{\alpha _1r_1}\left[ (K-{\overline{P}}_1)r_1-Ku_1{\overline{P}}^2_1\right] , \end{aligned}$$

  and ${\overline{P}}_1$ is the root of the polynomial $\displaystyle A_0\xi ^4+A_1\xi ^3+A_2\xi ^2+A_3\xi +A_4=0$, where $\displaystyle A_{0}=\frac{K^2u^2_{1}u_{2}}{\alpha ^2_{1}r^2_{1}}$, $\displaystyle A_{1}=\frac{2Ku_{1}u_{2}}{\alpha ^2_{1}r_{1}}$, $\displaystyle A_{2}=\frac{1}{\alpha ^2_{1}r_{1}}\left[ (r_{1}-2K^2u_{1})u_{2}-\alpha _{1}r_{2}u_{1}\right]$, $\displaystyle A_{3}=\frac{1}{\alpha ^2_{1}K}\left[ (\alpha _{1}\alpha _{2}-1)\alpha _{1}r_{2}-2K^2u_{2}\right]$, $\displaystyle A_{4}=\frac{1}{\alpha ^2_{1}}\left[ (1-\alpha _{1})\alpha _1r_2+K^2u_2\right]$. Since $A_{0},\ A_{1}>0$, the polynomial has a unique positive root if $A_2,\ A_3,\ A_4<0$, i.e., if

  $$\begin{aligned} \max \Big\{\frac{\alpha _1r_2}{2u_2}\left( \alpha _1\alpha _2-1\right) , \frac{1}{2u_1u_2}\left( r_1u_2-\alpha _1r_2u_1\right) \Big\}< K^2< \frac{\alpha _1r_2}{u_2}\left( \alpha _1-1\right) . \end{aligned}$$

• 5. The TPP-free equilibrium $E_4=({\check{P}}_1,0,{\check{Z}})$, where ${\check{P}}_1$ and ${\check{Z}}$ are positive solutions of the following nullclines:

  $$\begin{aligned} r_1\left( 1-\frac{P_1}{K}\right) -\frac{\beta _1(1-mZ)Z}{a_1+(1-mZ)P_1}-u_1P^2_1= & {} 0,, \end{aligned}$$

  (4)
  $$\begin{aligned} \frac{\lambda _1\beta _1(1-mZ)P_1}{a_1+(1-mZ)P_1}-\mu -\nu Z= & {} 0. \end{aligned}$$

  (5)

  From equation (4), we note the following:

  • (a) At $P_1=0$, we get the following quadratic equation in Z:

    $$\begin{aligned} \beta _1mZ^2-\beta _1Z+r_1a_1=0, \end{aligned}$$

    which has two positive real roots.

  • (b) At $Z=0$, we get the following quadratic equation in $P_1$:

    $$\begin{aligned} u_1KP^2_1+r_1P_1-r_1K=0, \end{aligned}$$

    which has a positive ($P^1_1$) and a negative ($P^2_1$) real roots.

  • (c) $\displaystyle \frac{dP_1}{dZ}=\frac{{\widetilde{A}}}{{\widetilde{B}}},$ where

    $$\begin{aligned} {\widetilde{A}}= & {} \frac{\beta (1-2mZ)}{a_1+(1-mZ)P_1}+\frac{\beta _1m(1-mZ)ZP_1}{(a_1+(1-mz)P_1)^2},\\ {\widetilde{B}}= & {} \frac{\beta _1(1-mZ)^2Z}{(a_1+(1-mZ)P_1)^2}-r_1-2u_1P_1.\\ \end{aligned}$$

    Thus, the derivative is negative if ${\widetilde{A}}$ and ${\widetilde{B}}$ are of the opposite signs.

  Similarly, from equation (5), we note the following:

  • (a) At $P_1=0$, we get $Z=-\frac{\mu }{v}<0$.

  • (b) At $Z=0$, we get $\displaystyle P_1=\frac{\mu a_1}{\lambda _1\beta _1-\mu }$, which is positive if $\lambda _1\beta _1>\mu$.

  • (c) $\displaystyle \frac{dP_1}{dZ}=\frac{\widetilde{A_1}}{\widetilde{B_1}},$ where

    $$\begin{aligned} \widetilde{A_1}= & {} \frac{\lambda \beta _1mP_1}{a_1+(1-mZ)P_1}-\frac{\lambda _1\beta _1m(1-mZ)P_1^2}{(a_1+(1-mz)P_1)^2}+v,\\ \widetilde{B_1}= & {} \frac{\lambda _1\beta _1(1-mZ)}{a_1+(1-mZ)P_1}-\frac{\lambda _1\beta _1m(1-mZ)^2P_1}{(a_1+(1-mz)P_1)^2}. \end{aligned}$$

    Thus, the derivative is positive if $\widetilde{A_1}$ and $\widetilde{B_1}$ are of the same sign. The above analysis shows that the nullclines (4) and (5) intersect uniquely in the interior of the first quadrant of the $P_1-Z$ plane if $\displaystyle \frac{\mu a_1}{\lambda _1\beta _1-\mu }<P^1_1$ and the quantities ${\widetilde{A}}$ and ${\widetilde{B}}$ are of the opposite signs whereas $\widetilde{A_1}$ and $\widetilde{B_1}$ are of the same sign.

• 6. The interior equilibrium $E^*=(P^*_1,P^*_2,Z^*)$, where

  $$\begin{aligned} Z^*=\frac{a_2+{P^*}^2_2}{\beta _2}\left[ r_2\left( 1-\frac{P^*_2+\alpha _2P^*_1}{K}\right) -u_2{P^*_2}^2\right] =f(P^*_1,P^*_2), \end{aligned}$$

  (6)

  $P^*_1$ and $P^*_2$ are positive solutions of the following nullclines:

  $$\begin{aligned} r_1\left( 1-\frac{P_1+\alpha _1P_2}{K}\right) -\frac{\beta _1(1-mf(P_1,P_2))f(P_1,P_2)}{a_1+(1-mf(P_1,P_2))P_1}-u_1P^2_1= & {} 0, \end{aligned}$$

  (7)
  $$\begin{aligned} \frac{\lambda _1\beta _1(1-mf(P_1,P_2))P_1}{a_1+(1-mf(P_1,P_2))P_1}-\frac{\lambda _2\beta _2P_2}{a_2+P^2_2}-\mu -\nu f(P_1,P_2)= & {} 0. \end{aligned}$$

  (8)

  While analytical complexity prevents us from deriving the exact conditions for the existence of interior equilibrium, numerical simulation demonstrates its existence. The points in intersection represent the population densities of the species at this equilibrium state (Fig. 2).

Now, we discuss about the local stability of the feasible equilibria of the proposed system (1).

• 1. The equilibrium $E_0$ is always unstable.

• 2. The equilibrium $E_1$ is stable if the following condition holds:

  $$\begin{aligned} \frac{K}{\alpha _2}<{\widehat{P}}_1 <\frac{a_1\mu }{\lambda _1\beta _1-\mu }. \end{aligned}$$

  (9)

  The analysis reveals that the system will reach a stable steady state where only the non-toxic phytoplankton exist if the density of the species falls within a specific range. One of the boundaries of the range depends on the competition coefficient $\alpha _2$. As $\alpha _2$ increases, indicating strong competition against toxic phytoplankton, the toxic phytoplankton is likely to be suppressed, potentially leading to its extinction. This encapsulates the survivability of the non-toxic species in the absence of its competitor. This biological phenomenon is reflected in the mathematical expression for the stability condition of $E_1$. However, the other boundary depends on the growth dynamics of zooplankton. In the absence of toxic species, non-toxic phytoplankton and zooplankton can coexist. To exclude this possibility, the net growth rate of zooplankton relative to its death rate, i.e., $\displaystyle \frac{\lambda _{1}\beta _{1}-\mu }{\mu }$ needs to be low.

• 3. The equilibrium $E_2$ is stable if the following condition is satisfied:

  $$\begin{aligned} \alpha _1>\frac{K}{\widetilde{P_2}}. \end{aligned}$$

  (10)

  The presence of toxic phytoplankton can lead to the eradication of non-toxic phytoplankton and zooplankton over time. This threat can arise from the intense competition for resources exerted by the toxic species. This can be mathematically described by a threshold level $\displaystyle K/\widetilde{P_2}$ for the competition coefficient $\alpha _1$. When this threshold is exceeded, the competition pressure becomes too strong for non-toxic species to survive, leading to their extinction. In the NTP-free scenario, the stability of the equilibrium $E_2$ becomes independent of the growth dynamics of zooplankton. This is because the long-term survivability of zooplankton depends solely on the availability of non-toxic food.

• 4. The equilibrium $E_3$, if exists, is stable provided,

  $$\begin{aligned} \frac{\lambda _1\beta _1{\overline{P}}_1}{a_1+{\overline{P}}_1}<\mu + \frac{\lambda _2\beta _2{\overline{P}}_2}{a_2+{\overline{P}}^2_2},\ 2K(r_2u_1{\overline{P}}_1+r_1u_2{\overline{P}}_2+2Ku_1u_2{\overline{P}}_1{\overline{P}}_2)>r_1r_2(\alpha _1\alpha _2-1). \end{aligned}$$

  (11)

  Therefore, for non-toxic and toxic phytoplankton to coexist in the absence of their potential predator, the total loss of zooplankton, which includes natural mortality ($\mu$) and mortality from consuming toxic phytoplankton $\displaystyle (\lambda _2\beta _2{\overline{P}}_2/\{a_2+{\overline{P}}^2_2\})$, must exceed their net growth through consumption of non-toxic resource. Interestingly, if the competition between two phytoplankton types is relatively weak ($\alpha _1 \alpha _2<1$), the second condition for stability becomes redundant. However, for intense competition ($\alpha _1 \alpha _2>1$), the second condition acts as a balancing mechanism between the equilibrium densities of the two phytoplankton species.

• 5. The equilibrium $E_4$, if exists, is stable if the following conditions are satisfied:

  $$\begin{aligned} \beta _2>\frac{a_2(K-\alpha _2{\check{P}}_1)r_2}{K{\check{Z}}}, \ B_1>0, \ B_2>0, \end{aligned}$$

  (12)

  where $B_1, B_2$ are defined in the proof.

  Therefore, in order to maintain a stable toxic phytoplankton-free equilibrium, the rate at which zooplankton consume toxic phytoplankton ($\beta _2$) must be greater than a certain factor times its intrinsic growth rate ($r_2$), i.e., the predation pressure on toxic phytoplankton should exceed its proliferation rate. This factor depends strongly on the equilibrium densities of existing non-toxic species and zooplankton.

• 6. The equilibrium $E^*$, if exists, is locally asymptotically stable if and only if the following conditions are satisfied:

  $$\begin{aligned} C_1>0, \ C_3>0, \ C_1C_2-C_3>0, \end{aligned}$$

  (13)

  where $C_i$’s ($i=1,2,3$) are defined in the proof.

The detailed proof is presented in Appendix 1.

Figure 2

Non-toxic species nullclines (solid curves) and zooplankton nullclines (dashed curves) for different levels of carrying capacity: (a) $K=3$, (b) $K=10$, (c) $K=15$.

The forced model

Environmental factors such as water temperature, light, thermocline depth, salinity, etc., are often important to varying degrees depending upon the ecological system. Thus, the parameters are subject to fluctuate with environmental changes³⁰. We consider the coefficient of refuge by non-toxic phytoplankton (m) in system (1) as periodically varying time-dependent function. Sinusoidal perturbation is assumed for the parameter under consideration. Thus the seasonality is superimposed as follows:

$$\begin{aligned} m(t)=m+m_1\sin (\omega t), \end{aligned}$$

(14)

where m is the average value of m(t). Here, $\omega$ represents the angular frequency of the fluctuation caused by seasonality.

Using this time dependent form of the refuge parameter (m) in system (1), the resulting forced plankton system takes the following form:

$$\begin{aligned} \frac{dP_1}{dt}= & {} r_1P_1\left( 1-\frac{P_1+\alpha _1 P_2}{K}\right) -\frac{\beta _1(1-m(t)Z)P_1Z}{a_1+(1-m(t)Z)P_1}-u_1P^3_1,\nonumber \\ \frac{dP_2}{dt}= & {} r_2P_2\left( 1-\frac{P_2+\alpha _2 P_1}{K}\right) -\frac{\beta _2P_2Z}{a_2+P^2_2}-u_2P^3_2,\nonumber \\ \frac{dZ}{dt}= & {} \frac{\lambda _1 \beta _1(1-m(t)Z)P_1Z}{a_1+(1-m(t)Z)P_1}-\frac{\lambda _2 \beta _2P_2Z}{a_2+P^2_2}-\mu Z-vZ^2. \end{aligned}$$

(15)

Plankton blooms can be predictably recurrent. The ecosystem not only experiences such events in the springtime but also repeats, over the winter and fall. Plankton population cycles are driven by the seasonal variation of physical conditions in the aquatic environment.

Table 1 Significations of the parameters in system (1) and the values employed for numerical simulations.

Numerical results

Here, we report extensive numerical simulations to further reveal the dynamical features of the autonomous system (1) and the associated forced system (15). We draw the phase portrait diagrams and time series solutions of the systems (1) and (15) using MATLAB’s ode45 solver. To track changes in the steady state behaviors of interacting populations, we perform several one-parameter bifurcation diagrams using the MatCont 6p11³² in MATLAB software.

Parameter estimation

Figure 3

Plots of the fitted model solution and the observed zooplankton density in Talsari (Orissa, India) and Digha Mohana (West Bengal) during the period January 12, 2000 to April 21, 2000.

We now proceed to determine the ‘best-fit’ estimates for certain model parameters through the application of the Least Square Method, in accordance with the observed behavior of field data. Water bodies in the Bay of Bengal experience eutrophication due to natural and human-induced sources, resulting in localized and intense blooms. The intensity of these planktonic blooms exhibits variation from one year to another. To analyze this phenomenon, we utilize real data sourced from³³. The data comprises plankton samples collected from Talsari (Orissa, India) and Digha Mohana (West Bengal, India), situated in the northeastern region of the Bay of Bengal. We have considered the data from January 12, 2000 to April 21, 2000. The intrinsic growth rate and refuge pattern of non-toxic phytoplankton have significant controlling commands on zooplankton and act as regulatory factors on plankton dynamics. We start with a set of initial parametric values (say, $p_a$) chosen from Table 1 except $r_2=0.1, K=12, \beta _1=0.8, \mu =0.025$, and estimate two significant parameters ${\widehat{\theta }}=(r_1, m)$. We choose the first-observed data point, $X(t=0)_{obs}=(P_1(t=0)_{obs}, P_2(t=0)_{obs}, Z(t=0)_{obs})$ as the initial population size. Subsequently, we proceed to perform a numerical solution of system (1) using the initial parameter values. We determine the solutions of the system at the time points corresponding to the availability of field data. These solutions are denoted as $X(t, p_a)_{mod}=(P_1(t)_{mod}, P_2(t)_{mod}, Z(t)_{mod})$, while the observed data points are represented as $X(t)_{obs}=(P_1(t)_{obs}, P_2(t)_{obs}, Z(t)_{obs})$. Now, we minimize the following cost function over the parametric space to estimate the parameters from the data points:

$$\begin{aligned} L(p_a|X_{obs})=\sum ^{t_{max}}_{t=t_0}\Vert X(t)_{obs}-X(t, p_a)\Vert ^2, \end{aligned}$$

where

$$\begin{aligned}{} & {} \Vert X(t)_{obs}-X(t, p_a)\Vert \\{} & {} \quad = \sqrt{(P_1(t)_{obs}-P_1(t)_{mod})^2+(P_2(t)_{obs}-P_2(t)_{mod})^2+(Z(t)_{obs}-Z(t)_{mod})^2}. \end{aligned}$$

The model fitting is presented in Fig. 3. The estimated values of the parameters are obtained as, $r_1=0.055$ and $m=0.55$. Therefore, the estimated growth rate of the non-toxic species of phytoplankton is low whereas the amount of refuge taken by the non-toxic phytoplankton is moderate. From the figure, we observe that the fitted model solution is in good agreement with the observed zooplankton density. Thus, our model reflects the behavior of a real ecosystem up to some extent.

Simulation results of system (1)

Figure 4

Effects of NTP uptake rate by zooplankton ($\beta _1$) and NTP refuge (m) on the equilibrium values of all the variables of the system (1). Rest of the parameters are at default levels.

Now, we see the effects of NTP refuge and NTP uptake rate by zooplankton on the equilibrium abundances of all the population densities for the system (1), Fig. 4. We observe that in the absence of both of these parameters, NTP and TPP are at higher equilibrium levels, while zooplankton vanishes due to the unavailability of NTP and toxicity of TPP. If the values of $\beta _1$ are increased, all the populations start to coexist in a stable mode. For higher values of $\beta _1$, zooplankton is at higher density, and consequently TPP population wipes out from the system. In this situation, the refuge of NTP brings back the coexistence scenario in the system.

From the bifurcation diagram in Fig. 5, we observe that for low values of $r_1$, the system showcases oscillatory coexistence around the unique interior equilibrium (region VIII). However, when the intrinsic growth rate of non-toxic species crosses the threshold $r_1=0.384$, a stable steady state devoid of toxic species emerges. Within the parameter window $0.384<r_1<0.425$, the system might either exhibit population cycles or becomes completely free of toxic phytoplankton, depending upon initial condition. This is because a higher growth rate of non-toxic species leads to a denser zooplankton population due to increased food availability. The combined effect of strong predation pressure from zooplankton and competition from the more abundant non-toxic species can drive the toxic phytoplankton to extinction, especially if their initial density is low. The instance when $r_1$ exceeds the threshold $r_1=0.425$, two pairs of interior equilibria are generated. The emergence of a stable coexistence steady state at this tipping point adds complexity to the system. As $r_1$ further increases, a situation of tristability emerges, with the system potentially settling into population cycles, steady state coexistence, or a toxic phytoplankton free equilibrium (region X). In this region, with as escalation in $r_1$, the densities of both non-toxic phytoplankton and zooplankton at the stable interior equilibrium increase, while the density of toxic species density continues to decline. Additionally, at $r_1 = 0.53$, two interior equilibria collide and mutually cease to exist. In region XI, the system exhibits similar bistable dynamics as observe in region IX. As $r_1$ crosses the threshold $r_1=0.684$, the system regains stability through Hopf bifurcation around the interior equilibrium $E^*$ with highest TPP abundance. The region XII depicts bistability between steady state coexistence and toxic species extinction. Moreover, at $r_1=0.728$, the system undergoes a regime shift, leading to a catastrophic collapse of the toxic phytoplankton. Region XII signifies the settlement of the system into environment free of toxic phytoplankton, irrespective of initial species densities. This implies that a high enough growth rate for non-toxic phytoplankton poses the risk of extinction to its competitor. As the system transitions from coexistence to eradication, it undergoes multiple regime shifts, becoming vulnerable to even small perturbations.

Figure 6a shows stable dynamics of the system (1) around the interior equilibrium $E^*$ for $K=3$ and a Hopf bifurcation in the form of limit cycle is illustrated in Fig. 6(b) for $K=5.85$. We determine minima and maxima of population cycles by simulating the system over a span of 20, 000 time steps. To ensure that transient dynamics are excluded, we analyze the behavior of the system during the final 15, 000 time steps. We infer from Fig. 7(a) that for lower values of $\beta _2$, all the populations coexist in a stable mode, while on increasing it, we obtain three critical values of $\beta _2$, namely $\beta ^1_2=0.4298$, $\beta ^2_2=0.4534$ and $\beta ^3_2=0.4708$ such that at $\beta _2=\beta ^1_2$ the system bifurcates through a supercritical Hopf bifurcation and generates periodic orbits. If the values of $\beta _2$ are further increased, the cycles disappear via subcritical Hopf bifurcation at $\beta _2=\beta ^2_2$. Interestingly, at the threshold $\beta _2=\beta ^3_2$, the TPP-free steady stable emanates from the former, which shows the occurrence of a transcritical bifurcation between the equilibria $E^*$ and $E_4$. Ecologically, an increase in $\beta _2$ leads to a rise in the equilibrium density of zooplankton. This, in turn, strengthens predation pressure on phytoplankton. Consequently, the equilibrium densities of both phytoplankton decrease, as shown by the descending equilibrium curve. However, within a specific range of $\beta$, the bloom phenomenon is evident. When predation pressure becomes high, the defense mechanism of toxic phytoplankton becomes ineffective, leading to their eradication. In this scenario, the ecosystem reaches a state where non-toxic phytoplankton and zooplankton coexist. Fig. 7(b) depicts that the system produces limit cycle oscillations for lower values of $u_1$, and settles to the stable equilibrium position $E^*$ through a subcritical Hopf bifurcation for higher levels of the coefficient of toxicity. This shows that increased toxicity caused by anthropogenic activities disrupts the conditions necessary for recurrent blooms.

Now, we study the stability structure of the proposed system in presence of multiple coexisting steady states with respect to the refuge parameter m (Fig. 8). Here, blue curve and cyan line represent the stable branches, while red curve and black line represent the unstable branches of the interior equilibrium $E^*$ and the TPP-free equilibrium $E_4$, respectively; green curve indicates the unstable zooplankton-free steady state $E_3$. We observe that for $m<0.092$, the system possesses a unique interior equilibrium, which is unstable. The system exhibits periodic cycles in this region (not captured in the figure). The system is bistable in the region II. Depending on initial species densities, either population cycles or extinction of toxic species is observed. Therefore, for low refuge of non-toxic phytoplankton, high predation pressure on toxic phytoplankton leads to their eradication. At the threshold $m=0.099$, the system undergoes a tipping event, resulting in the abrupt emergence of toxic phytoplankton and the coexistence of both phytoplankton and zooplankton. Consequently, the refuge by non-toxic phytoplankton establishes the possibility of the long-term survival of its competitor at a steady state. As m increases further, the reduced availability of basal food causes a decline in zooplankton density, which in turn allows both phytoplankton to thrive. For $0.099 \le m \le 0.1268$, tristability among cyclic coexistence, steady state coexistence, and toxic phytoplankton-free scenario. More precisely, if the initial density of toxic phytoplankton is high, they can effectively defend against zooplankton and compete for resources with non-toxic phytoplankton, ensuring their long-term survival. However, if their initial density is low, their survival is compromised. At $m=0.1268$, population cycles disappear and the system switches from instability to stability at the interior equilibrium with highest TPP abundance. In the tristable region IV, the system possesses a pair of interior attractors and the boundary attractor $E_4$. Therefore, depending on the initial species densities, either all species coexist or toxic phytoplankton die out. Additionally, within this range of refuge, initial conditions play a pivotal role in determining the equilibrium densities at the interior state, resulting in either high or low densities. Simulation results depict the emergence of another coexistence scenario via a transcritical bifurcation at $m=0.1438$. This pivotal bifurcation prevents the extinction of toxic phytoplankton. Therefore, a strong refuge capacity of non-toxic phytoplankton ensures the thriving of its competitor. The region V represents tristability at three possible interior steady states for $0.1438 \le m \le 0.1442$. Above the threshold level $m=0.1442$ two coexisting equilibria of opposite stability nature annihilate each other, yielding a saddle-node bifurcation. The bistable region VI determines a pair of coexisting attractors, for different initial densities. For $m > 0.1464$, the region VII encapsulates that the system exhibits monostability around the unique interior equilibrium $E^*$. Therefore, the refuge by non-toxic species enhances the resilience of the plankton system, eliminates various complex multistable scenarios, and promotes a stable environment where all plankton species coexist.

Figure 5

Bifurcation diagram of system (1) with respect to $r_1$ for $K=12$.

Figure 6

Phase portraits of system (1) with the default parametric levels except in (a) $K=3$, and (b) $K=5.85$.

Figure 7

Bifurcation diagram of system (1) with respect to (a) $\beta _2$ for $\lambda _2=0.2$, and (b) $u_1$ for $K=12$, $r_2=0.3$ and $u_2=0.5$. Here, blue and red curves respectively represent stable and unstable modes around the interior equilibrium; black dots represent the minimal and maximal amplitudes of the plankton biomass; cyan line represents stable steady state around the TPP-free equilibrium.

Figure 8

Bifurcation diagram of system (1) with respect to m for $K=12$.

Figure 9

Bifurcation diagram of system (1) with respect to $r_2$ for $K=12$.

The bifurcation diagram with respect to $r_2$ shows that the system experiences stable dynamics in the region XIII around the toxic phytoplankton-free steady state $E_4$ (Fig. 9). At the tipping point $r_2=0.4982$, the system undergoes a critical transition marked by the sudden establishment of coexistence among the interacting species. The elevated growth increases their density, enhancing their compete ability and defensive strategy. This potential improvement is reflected in the increasing equilibrium curve of the toxic species. For $0.4982<r_2<0.705$, bistability between the equilibria $E^*$ and $E_4$ is observed. Therefore, the long-term survival of toxic phytoplankton depends on initial densities. At $r_2=0.705$, the equilibrium $E_4$ loses stability and another interior equilibrium emerges via a smooth transition. This suggests that exceeding this threshold completely eliminates the risk of extinction of toxic phytoplankton. In the narrow parameter window $0.705<r_2<0.7065$, the system might settle into either an interior equilibrium with high toxic phytoplankton density or one with low density. Interestingly, the possibility of coexistence with low toxic phytoplankton density disappears abruptly at another tipping point $r_2=0.7065$. Beyond this threshold, the system undergoes a pivotal regime shift and settles into the coexistence state. Overall, the increase in growth rate leads to a sudden emergence of toxic phytoplankton, but their escape from the risk of extinction is a non-catastrophic event.

Figure 10

Bifurcation diagram of system (1) with respect to K.

We observe that for low carrying capacity of the environment, all interacting species coexist in a stable steady state (Fig. 10). Interestingly, at the threshold level $K=3.8$, the risk of extinction of toxic phytoplankton appears. If the initial density of toxic phytoplankton is very low compared to non-toxic species, they cannot compete in the long run, and the environment will only support the non-toxic species. Once the carrying capacity surpasses the threshold $K=5.72$, the stable configuration at the coexistence state is disrupted, leading to fluctuations in population levels. This cyclical coexistence persists as the carrying capacity increases further. At the threshold $K=11.71$, a stable coexistence pattern suddenly reappears. As the carrying capacity continues to improve, the population levels of all species increase due to the healthier environment. However, beyond this threshold, the coexistence pattern (steady or oscillatory) depends on initial conditions. Importantly, the risk of extinction for the toxic species still exists.

Figure 11

Transcritical bifuraction diagram of system (1) with respect to $r_1$. Here, red color denotes stable interior equilibrium $E^*$, green color represents stable TPP-free equilibrium $E_4$ and the black dot stands for the bifurcation point.

Fig. 11 shows the occurrence of a transcritical bifurcation between the equilibria $E^*$ and $E_4$. We vary the intrinsic growth rate of NTP ($r_1$) in the interval [0.755, 0.77] and plot the density of TPP against $r_1$. We note that when the parameter $r_1$ has low values, the coexisting equilibrium $E^*$ remains stable. However, as the value of $r_1$ surpasses the critical threshold of $r^*_1 = 0.762$, the stability of equilibrium $E^*$ is lost. Beyond this point, a stable TPP-free equilibrium $E_4$ emerges from the previously mentioned equilibrium.

The above bifurcation analyses clearly indicate that the proposed system exhibits multiple interior equilibria depending upon the levels of model parameters. For instance, the system possesses at most six such equilibria for different levels of m and K. For a better visualization, the bi-parametric space (m, K) is partitioned into different sub-regions depending on the number of interior equilibria (Fig. 12).

Figure 12

The number of interior equilibria of system (1) in (m, K) plane. Here, blue, red, green, cyan, magenta and black colors represent the existence of one, two, three, four, five and six interior equilibria, respectively.

Figure 13

System (15) shows periodic behavior for $m(t)=m+m_1\sin (\omega t)$ for $m=0.3$ and $m_1=0.05$.

Simulation results of system (15)

Figure 14

System (15) shows double periodic behavior for $m(t)=m+m_1\sin (\omega t)$ for $m=0.3$ and $m_1=0.25$.

Figure 15

System (15) shows chaotic dynamics for $m(t)=m+m_1\sin (\omega t)$ for $K=12$, $m=0.05$ and $m_1=0.035$.

Figure 16

System (15) shows almost periodic behavior for $m(t)=m+m_1[\sin (\omega t)+\cos (\sqrt{\omega })t]$ for $m=0.3$ and $m_1=0.05$.

We now conduct numerical simulations to examine the dynamic characteristics of the forced system (15). To account for seasonal changes in the refuge pattern, we introduce a time-dependent element by expressing m(t) as a periodic function. Specifically, we model this parameter using a sinusoidal function, $m(t)=m+m_1\sin (\omega t)$, with $\omega =2\pi /365$. Analytically, we represent this forcing as

$$\begin{aligned} m(t)={\left\{ \begin{array}{ll} m+m_{1}, \ \text {High season}\\ m-m_{1}, \ \text {Low season} \end{array}\right. } \end{aligned}$$

Here, $m_1$ ($0<m_1<m$) controls the strength of forcing.

Our observations reveal that the forced system (15) generates a positive periodic solution (Fig. 13). In the scenario where the system is in a stable steady state, the seasonal fluctuations in refuge result in the emergence of a periodic solution (depicted in Fig. 13) and even a 2-cycle periodic solution (displayed in Fig. 14). However, seasonal changes in refuge add complexity to an otherwise unstable system by producing clusters of irregular oscillations (see Fig. 15). Such chaotic nature of the forced system can be explained through incommensurable frequencies of limit cycles³⁴. To account for the almost periodic nature of the temporally varying environment, we model the parameter m(t) as an almost periodic function of time: $m(t)=m+m_1[\sin (\omega t)+\cos (\sqrt{\omega })t]$³⁵. This choice results in an almost periodic coexistence scenario, as illustrated in Figure 16.

To verify the chaotic behavior of system (15), we compute the maximum Lyapunov exponent using the Wolf algorithm³⁶. The maximum Lyapunov exponent corresponding to the scenario depicted in Figure 15 is presented in Figure 17(a). The positive value of the maximum Lyapunov exponent confirms the chaotic dynamics of the system. Figure 17(b) showcases the Poincaré section of the chaotic attractor on the $P_1 = 1.2$ plane. The scattered distribution of sampling points in the section signifies the chaotic nature of the system.

Figure 17

(a) The maximum Lyapunov exponent of the system (15); (b) Poincaré section in the $P_2-Z$ plane when $P_1$ is fixed at 1.2. Parameters are at the same levels as in Fig. 15.

Conclusion

Plankton in aquatic environments exhibit fascinating anti-predator behaviors. Toxic phytoplankton deploy a chemical defense, releasing toxins to deter predators. In contrast, non-toxic species often resort to hiding behaviors to avoid detection. These defensive strategies extend to zooplankton, which can actively avoid dense patches of harmful phytoplankton. While the role of toxins in controlling plankton blooms is well-established, the combined impact of these contrasting defensive strategies on bloom dynamics, especially in polluted environments, remains unexplored. Motivated by this, we present a mathematical framework delineating the dynamic interplay among non-toxic phytoplankton, toxic phytoplankton, and zooplankton species. The growth of zooplankton is assumed to depend solely upon the non-toxic phytoplankton. The introduction of Monod-Haldane response function to represent the grazing phenomenon of toxic phytoplankton by zooplankton encapsulates the fact that the uptake rate of the toxic species decreases with the increase of their biomass. As a consequence, the negative feedback of zooplankton on toxic phytoplankton becomes less significant when they are abundant. Non-toxic phytoplankton take density-dependent refuge to prolong their survival. Also, the anthropogenic contaminants released into the aquatic system possess harmful impacts on both phytoplankton as well as zooplankton.

Our proposed model is fitted to the field data from Talsari and Digha Mohana, India to manifest the resemblance with a real marine ecosystem. We estimate the intrinsic growth rate ($r_1$) and coefficient of refuge (m) of non-toxin phytoplankton consistent with the behavior of the real data. The outcomes of our simulations reveal that nutrient enrichment plays a disruptive role in the dynamics of the system, aligning with the contentious predictions referred to as the paradox of enrichment³⁷. This finding substantiates the argument that eutrophic environmental conditions exert a significant influence on the initiation of phytoplankton blooms. Intriguingly, nutrient enrichment does not always lead to the expected outcome of bloom formation. Our investigation suggests a more nuanced scenario, where initial species densities can influence the system towards a steady state coexistence or even the unexpected collapse of toxic phytoplankton. To diminish the risk, the availability of huge quantities of suitable nutrients for phytoplankton must be controlled. The refuge adopted by the non-toxic counterpart holds the capacity to manage the escalation of algal proliferation by eliminating the presence of population cycles. From ecological perspective, this result supports that the refuge capacity can lead to bloom termination. A similar stabilizing effect of phytoplankton refuge on plankton population dynamics is also observed by Mandal et al.¹¹ However, our findings differ from previous studies by demonstrating the existence of multiple stable states. Therefore, initial plankton densities play a crucial role in determining the fate of the plankton community. If initial densities are conducive, plankton populations may undergo seasonal fluctuations, with bloom followed by decline with increasing refuge capacity. Interestingly, with the same refuge size, depending upon initial densities, plankton populations might reach a stable equilibrium and maintain constant densities. Some extreme scenarios can also be observed: if the initial density of toxic phytoplankton is low, the refuge may hinder the long-term survival of the toxic species under identical environmental conditions. With increasing refuge capacities the non-toxic food becomes less available to zooplankton, which in turn drives the zooplankton density to a low level. As a consequence, the toxic species can survive with a higher density. Previously, Li et al.²⁵ investigated the effects of phytoplankton refuge and toxicity on the dynamics of a phytoplankton-zooplankton system. They showed that when refuge capacity is very small, the periodic phenomenon goes away with the increase of phytoplankton refuge; for elevated refuge capacity, the system exhibits switching dynamics. However, our study captures more complicated dynamical scenarios with varying capacities of density-dependent refuge. We observe the phenomenon of tristability among three coexistence steady states in a narrow range of the refuge coefficient. This finding is novel within the realm of plankton ecology. Different types of multistability phenomena underscore the heightened susceptibility of the plankton model to minor disturbances. Anthropogenic toxicants may also curtail population variability by preventing species from fluctuating around a coexisting equilibrium. Our result on anthropogenic toxicants is consistent with the findings of Chakraborty and Das¹². Our finding documents that uptake of TPP by grazer zooplankton triggers the occurrence of stability switches within the system. From a biological perspective, this outcome substantiates the notion that the onset or cessation of typical blooms hinges on the zooplankton’s uptake rate of toxic species.

The intrinsic growth rates of both non-toxic and toxic phytoplankton also stimulate the emergence of multistability scenarios in the system. A surge in the growth rate of non-toxic species can trigger a catastrophic collapse of its competitor, shattering a healthy coexistence. This transition from a balanced state to eradication involves multiple regime shifts, affecting the resilience of the system. Interestingly, when the growth rate of toxic species increases, it can rebound from this crash, re-establishing a coexistence. This coexistence appears to be oscillatory. A higher growth rate triggers a sudden steady state coexistence. This findings highlights the crucial role of initial species densities in shaping the ultimate coexistence pattern.

Numerous environmental elements, including water temperature, light intensity, and rainfall, exhibit periodic seasonal variations. These variations consequently instigate seasonal influences on the refuge behavior of phytoplankton. With this consideration, we delved into a seasonally forced model aligned with the proposed autonomous model. Our model exhibits a diverse range of dynamic characteristics, spanning from straightforward cyclic blooms to intricate chaotic bloom occurrences. The cyclic solution behavior indicates the development of predictably recurrent plankton bloom events. Increased seasonal strength gives rise to multi-annual cycles within an otherwise stable system. Furthermore, when the unforced system displays sustained oscillations, seasonal fluctuations in the refuge pattern generate clusters of irregular amplitude oscillations. Therefore, a variation in the strength of seasonality in NTP refuge pattern changes the nature of the plankton bloom. Ecologically, a higher strength of seasonality stimulates the occurrence of irregular bloom, which showcases the unpredictability of bloom event. In the context of real-world observations, Medvinsky and colleagues³⁸ studied phytoplankton and zooplankton data from Naroch Lakes, Belarus, collected from 1993 to 2013. Samples were collected monthly at monitoring points during the vegetative season (from May to October). They employed advanced statistical methods, such as recurrence quantification analysis, to examine the observed irregular fluctuations in plankton abundances. Their findings demonstrated that chaos, even far away from the edge of chaos, can indeed occur in aquatic communities. Previously, Mandal et al.¹¹ studied a two-tiered phytoplankton-zooplankton system where the refuge depends solely on phytoplankton density. Their findings showed that variations in the strength of seasonal forcing on refuge led to the emergence of periodic, 2-cycle periodic, or even 3-cycle periodic solutions.

In conclusion, the findings acquired will equip resource managers with essential insights for devising efficient control strategies aimed at managing and mitigating blooms, along with their often disastrous impacts on coastal ecosystems.