# Stability and Bifurcations in a Nutrient–Phytoplankton–Zooplankton Model with Delayed Nutrient Recycling with Gamma Distribution

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## Abstract

**:**

## 1. Introduction

## 2. The Models

#### 2.1. The Weak Model

#### 2.2. The Strong Model

#### 2.3. The Model without Delay

## 3. Equilibrium Solutions

- -
- A trivial equilibrium ${E}_{1}$, with no phytoplankton and no zooplankton;
- -
- An equilibrium with phytoplankton and no zooplankton, denoted ${E}_{2}$;
- -
- An equilibrium with both phytoplankton and zooplankton, denoted ${E}_{3}$.

#### 3.1. Equilibrium Points for the System without Delay

#### 3.2. Equilibrium Points for the Reduced Weak System (7)

#### 3.3. Equilibria for the Reduced Strong Model (15)

- (1)
- The trivial equilibrium ${E}_{1}=\left({N}_{T},0,0,0\right),$ for all ${N}_{T}\ge 0;$
- (2)
- The equilibrium with no zooplankton ${E}_{2}=\left(\widehat{N},\widehat{P},0,{\widehat{Q}}_{1}\right),$ with ${Q}_{1}=\frac{\tau \lambda}{2}\widehat{P},$for all ${N}_{T}$, with ${N}_{T}\ge {N}_{{T}_{1}},$ if $\lambda <\mu $;
- (3)
- The equilibrium ${E}_{3}=\left({N}^{*},{P}^{*},{Z}^{*},{Q}_{1}^{*}\right)$, with ${Q}_{1}^{*}=\frac{\mu \tau}{2}f\left({N}^{*}\right){h}^{-1}\left(\frac{\delta}{\gamma g}\right),$for all ${N}_{T},$ with ${N}_{T}\ge {N}_{{T}_{2}}\left(\tau \right),$ if $\lambda <\mu $ and $\delta <\gamma g$.

- The equilibrium point ${E}_{1}$ is unaffected by the delay.
- For the equilibrium point ${E}_{2}$, the value of P is reduced by the delay.
- For the equilibrium point ${E}_{3}$, the values of N and Z are reduced by the delay.
- The first transition point is unaffected by the delay, ${N}_{{T}_{1}}={N}_{{T}_{1}}\left(\tau \right),$ while the second transition point is increased by the delay, ${N}_{{T}_{2}}<{N}_{{T}_{2}}\left(\tau \right),$ if $\tau >0.$

## 4. Local Stability

- for ${N}_{T}<{N}_{{T}_{1}},$ the only equilibrium point is ${E}_{1},$ and it is asymptotically stable,
- for ${N}_{{T}_{1}}<{N}_{T}<{N}_{{T}_{2}}\left(\tau \right)$ the equilibrium ${E}_{2}$ is asymptotically stable, while ${E}_{1}$ is unstable,
- and, finally, as ${N}_{T}>{N}_{{T}_{2}}\left(\tau \right),$ the equilibrium ${E}_{3}$ is asymptotically stable either for all ${N}_{T}>{N}_{{T}_{2}}\left(\tau \right)$ or there exists an ${N}_{{T}_{3}}\left(\tau \right)$ such that ${E}_{3}$ is asymptotically stable for ${N}_{{T}_{2}}\left(\tau \right)<{N}_{T}<{N}_{{T}_{3}}\left(\tau \right)$, and unstable for ${N}_{T}>{N}_{{T}_{3}}\left(\tau \right),$ depending on the response function $h,$ while the other two equilibria are unstable.

#### 4.1. The System without Delay

#### 4.2. The Weak Model Case

**Proposition**

**1.**

- (i)
- If ${N}_{T}<{N}_{{T}_{1}}$, then ${E}_{1}$ is locally asymptotically stable in ${D}_{1}$;
- (ii)
- If ${N}_{T}>{N}_{{T}_{1}}$, then ${E}_{1}$ is a (2,1) type saddle point;
- (iii)
- If ${N}_{T}={N}_{{T}_{1}}$, then ${E}_{1}$ is a fold singularity.

**Proof.**

**Proposition**

**2.**

- (i)
- if ${N}_{T}$ $={N}_{{T}_{1}}$ or ${N}_{T}={N}_{{T}_{2}}\left(\tau \right)$ then ${E}_{2}$ is a fold singularity;
- (ii)
- if ${N}_{T}>{N}_{{T}_{2}}\left(\tau \right)$ then ${E}_{2}$ is a saddle point of type (2,1);
- (iii)
- if ${N}_{T}$ $<{N}_{{T}_{1}}$ then ${E}_{2}$ is not in ${D}_{1}.$

**Proof.**

**Proposition**

**3.**

**Proof.**

**Proposition**

**4.**

- (i)
- For ${N}_{T}>{N}_{{T}_{2}}\left(\tau \right),$ close to ${N}_{{T}_{2}}\left(\tau \right),$ the equilibrium point ${E}_{3}$ is an attractor.
- (ii)
- If ${a}_{1}{a}_{2}-{a}_{3}>0$, then ${E}_{3}$ is locally asymptotically stable.
- (iii)
- If ${a}_{1}{a}_{2}-{a}_{3}=0$ then ${E}_{3}$ is a Hopf singularity.
- (iv)
- If ${a}_{1}{a}_{2}-{a}_{3}<0$ then ${E}_{3}$ is a (1,2) saddle point. In addition, for each τ there exists a value ${N}_{{T}_{3}}\left(\tau \right),$ given by$${N}_{{T}_{3}}\left(\tau \right)=min\left\{{N}_{T},\phantom{\rule{1.em}{0ex}}{N}_{T}>{N}_{{T}_{2}}\left(\tau \right),\phantom{\rule{1.em}{0ex}}{a}_{1}{a}_{2}-{a}_{3}=0\right\},$$

**Proof.**

#### 4.3. The Strong Model Case

**Proposition**

**5.**

- (i)
- If ${N}_{T}<{N}_{{T}_{1}}$, then ${E}_{1}$ is locally asymptotically stable in ${D}_{2}.$
- (ii)
- If ${N}_{T}>{N}_{{T}_{1}}$, then ${E}_{1}$ is a (3,1) type saddle point.
- (iii)
- If ${N}_{T}={N}_{{T}_{1}}$, then ${E}_{1}$ is a fold singularity.

**Proof.**

**Proposition**

**6.**

- (i)
- If ${N}_{T}$ $={N}_{{T}_{1}}$ or ${N}_{T}={N}_{{T}_{2}}\left(\tau \right)$, then the equilibrium ${E}_{2}$ is a fold singularity;
- (ii)
- If ${N}_{T}>{N}_{{T}_{2}}\left(\tau \right)$, then the equilibrium ${E}_{2}$ is a saddle point of type (3,1);
- (iii)
- If ${N}_{T}$ $<{N}_{{T}_{1}}$, then the equilibrium ${E}_{2}$ is not in ${D}_{2}.$

**Proof.**

**Proposition**

**7.**

- (i)
- For ${N}_{T}>{N}_{{T}_{2}}\left(\tau \right),$ close to ${N}_{{T}_{2}}\left(\tau \right),$ the equilibrium point ${E}_{3}$ is an attractor.
- (ii)
- If one of the conditions ${b}_{j}>0,$ $\phantom{\rule{0.166667em}{0ex}}j=\overline{1,6}$, in (44) is not satisfied, then ${E}_{3}$ is unstable. In addition, for each τ there exists a value ${N}_{{T}_{3}}\left(\tau \right),$ given by$${N}_{{T}_{3}}\left(\tau \right)=min\left\{{N}_{T}\left|{N}_{T}>{N}_{{T}_{2}}\left(\tau \right),\right.{\prod}_{j=1}^{6}{b}_{j}=0\right\},$$

**Proof.**

**Remark**

**1.**

**Proposition**

**8.**

- (i)
- If $\left({b}_{1}{b}_{2}-{b}_{3}\right){b}_{3}-{b}_{1}^{2}{b}_{4}>0,$ then the equilibrium ${E}_{3}$ of system (15) is locally asymptotically stable for all ${N}_{T}>{N}_{{T}_{2}}\left(\tau \right).$
- (ii)
- If $\left({b}_{1}{b}_{2}-{b}_{3}\right){b}_{3}-{b}_{1}^{2}{b}_{4}=0,$ then ${E}_{3}$ is a Hopf singularity.
- (iii)
- If $\left({b}_{1}{b}_{2}-{b}_{3}\right){b}_{3}-{b}_{1}^{2}{b}_{4}<0,$ then ${E}_{3}$ is unstable. In addition, for each τ, there exists a value ${N}_{{T}_{3}}\left(\tau \right),$ given by$${N}_{{T}_{3}}\left(\tau \right)=min\left\{{N}_{T}\left|{N}_{T}>{N}_{{T}_{2}}\left(\tau \right)\right.,\left({b}_{1}{b}_{2}-{b}_{3}\right){b}_{3}-{b}_{1}^{2}{b}_{4}=0\right\},$$

**Proof.**

## 5. Local Bifurcations

#### 5.1. Transcritical Bifurcations

- (i)
- at ${N}_{T}={N}_{{T}_{1}},$ the equilibrium points ${E}_{1}$ and ${E}_{2}$ collide and interchange stability;
- (ii)
- at ${N}_{T}={N}_{{T}_{2}}\left(\tau \right),$ the equilibrium points ${E}_{2}$ and ${E}_{3}$ collide and interchange stability.

#### 5.1.1. Transcritical Bifurcations for the Weak Model

**Proposition**

**9.**

**Proof.**

**Proposition**

**10.**

**Proof.**

**Remark**

**2.**

#### 5.1.2. Transcritical Bifurcations for the Strong Model

**Proposition**

**11.**

**Proof.**

**Proposition**

**12.**

**Proof.**

**Remark**

**3.**

#### 5.2. Hopf Bifurcations

#### 5.2.1. Hopf Bifurcations for the Weak Model

#### 5.2.2. Hopf Bifurcation for the Strong Model

**Remark**

**4.**

## 6. Discussion

- (1)
- For ${N}_{T}<{N}_{{T}_{1}},$ there is only one equilibrium point with no phytoplankton and no zooplankton (${E}_{1})$, which is asymptotically stable;
- (2)
- For ${N}_{{T}_{1}}<{N}_{T}<{N}_{{T}_{2}}\left(\tau \right)$ the equilibrium ${E}_{2}$ with phytoplankton and no zooplankton is asymptotically stable, while ${E}_{1}$ is unstable;
- (3)
- As ${N}_{T}>{N}_{{T}_{2}}\left(\tau \right),$ the first two equilibria are unstable, while the equilibrium ${E}_{3}$ with both phytoplankton and zooplankton is asymptotically stable either for all ${N}_{T}>{N}_{{T}_{2}}\left(\tau \right)$ or there exists an ${N}_{{T}_{3}}\left(\tau \right)$ such that ${E}_{3}$ is stable for ${N}_{{T}_{2}}\left(\tau \right)<{N}_{T}<{N}_{{T}_{3}}\left(\tau \right)$, and unstable for ${N}_{T}>{N}_{{T}_{3}}\left(\tau \right),$ close to ${N}_{{T}_{3}}\left(\tau \right),$ depending on the response function $h.$

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**$N,P,Z,Q$ as functions of ${N}_{T}$, for fixed $\tau =5$, for the equilibrium points ${E}_{1}$ (blue line), ${E}_{2}$ (green line), ${E}_{3}$ (red line), using a type II response.

**Figure 2.**$N,P,Z,{Q}_{1}$ as functions of ${N}_{T}$, for fixed $\tau =5$, for the equilibrium points ${E}_{1}$ (blue line), ${E}_{2}$ (green line), ${E}_{3}$ (red line), using a type III response.

**Figure 3.**Regions in the $(\tau ,{N}_{T})$ plane that exhibit different behaviours for the equilibrium ${E}_{3}$, using the Type II response for: (

**a**) the weak model; (

**b**) the strong model. Region 3 is where the ${E}_{3}$ and is stable, but where ${E}_{1}$, ${E}_{2}$ are unstable. For parameters on the curve separating regions 3 and 4, ${E}_{3}$ is a Hopf singularity, while in region 4, ${E}_{3}$ is unstable. A Hopf bifurcation may take place when parameters cross from region 3 to region 4.

**Figure 4.**Simulations for the weak model, using a type II response $h\left(P\right)=\frac{P}{P+{k}_{P}}$: (

**a**) $\tau =5$, ${N}_{T}=1.05$, showing an evolution towards ${E}_{3}$; (

**b**) $\tau =5$, ${N}_{T}=1.2$, showing a periodic behavior; (

**c**) projections of the attractor, for $\tau =5$, ${N}_{T}=1.2$, $t\in [400,500]$. The stable limit cycle may appear through a supercritical Hopf bifurcation at ${N}_{{T}_{3}}\left(\tau \right)=1.096$.

**Figure 5.**Simulations for the strong model, using a type II response $h\left(P\right)=\frac{P}{P+{k}_{P}}$. Projections of the attractor for $\tau =5$ and (

**a**) ${N}_{T}=1.05$; (

**b**) ${N}_{T}=1.096$; (

**c**) ${N}_{T}=1.2$; $t\in [700,800]$. The stable limit cycle may appear through a supercritical Hopf bifurcation at ${N}_{{T}_{3}}\left(\tau \right)=0.955$.

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**MDPI and ACS Style**

Sterpu, M.; Rocşoreanu, C.; Efrem, R.; Campbell, S.A.
Stability and Bifurcations in a Nutrient–Phytoplankton–Zooplankton Model with Delayed Nutrient Recycling with Gamma Distribution. *Mathematics* **2023**, *11*, 2911.
https://doi.org/10.3390/math11132911

**AMA Style**

Sterpu M, Rocşoreanu C, Efrem R, Campbell SA.
Stability and Bifurcations in a Nutrient–Phytoplankton–Zooplankton Model with Delayed Nutrient Recycling with Gamma Distribution. *Mathematics*. 2023; 11(13):2911.
https://doi.org/10.3390/math11132911

**Chicago/Turabian Style**

Sterpu, Mihaela, Carmen Rocşoreanu, Raluca Efrem, and Sue Ann Campbell.
2023. "Stability and Bifurcations in a Nutrient–Phytoplankton–Zooplankton Model with Delayed Nutrient Recycling with Gamma Distribution" *Mathematics* 11, no. 13: 2911.
https://doi.org/10.3390/math11132911