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Structured Model Conserving Biomass for the Size-spectrum Evolution in Aquatic Ecosystems

Laura Kanzler, Benoit Perthame, Benoit Sarels

Abstract

Mathematical modelling of the evolution of the size-spectrum dynamics in aquatic ecosystems was discovered to be a powerful tool to have a deeper insight into impacts of human- and environmental driven changes on the marine ecosystem. In this article we propose to investigate such dynamics by formulating and investigating a suitable model. The underlying process for these dynamics is given by predation events, causing both growth and death of individuals, while keeping the total biomass within the ecosystem constant. The main governing equation investigated is deterministic and non-local of quadratic type, coming from binary interactions. Predation is assumed to strongly depend on the ratio between a predator and its prey, which is distributed around a preferred feeding preference value. Existence of solutions is shown in dependence of the choice of the feeding preference function as well as the choice of the search exponent, a constant influencing the average volume in water an individual has to search until it finds prey. The equation admits a trivial steady state representing a died out ecosystem, as well as - depending on the parameterregime - steady states with gaps in the size spectrum, giving evidence to the well known cascade effect. The question of stability of these equilibria is considered, showing convergence to the trivial steady state in a certain range of parameters. These analytical observations are underlined by numerical simulations, with additionally exhibiting convergence to the non-trivial equilibrium for specific ranges of parameters.

Structured Model Conserving Biomass for the Size-spectrum Evolution in Aquatic Ecosystems

Abstract

Mathematical modelling of the evolution of the size-spectrum dynamics in aquatic ecosystems was discovered to be a powerful tool to have a deeper insight into impacts of human- and environmental driven changes on the marine ecosystem. In this article we propose to investigate such dynamics by formulating and investigating a suitable model. The underlying process for these dynamics is given by predation events, causing both growth and death of individuals, while keeping the total biomass within the ecosystem constant. The main governing equation investigated is deterministic and non-local of quadratic type, coming from binary interactions. Predation is assumed to strongly depend on the ratio between a predator and its prey, which is distributed around a preferred feeding preference value. Existence of solutions is shown in dependence of the choice of the feeding preference function as well as the choice of the search exponent, a constant influencing the average volume in water an individual has to search until it finds prey. The equation admits a trivial steady state representing a died out ecosystem, as well as - depending on the parameterregime - steady states with gaps in the size spectrum, giving evidence to the well known cascade effect. The question of stability of these equilibria is considered, showing convergence to the trivial steady state in a certain range of parameters. These analytical observations are underlined by numerical simulations, with additionally exhibiting convergence to the non-trivial equilibrium for specific ranges of parameters.
Paper Structure (22 sections, 3 theorems, 80 equations, 10 figures)

This paper contains 22 sections, 3 theorems, 80 equations, 10 figures.

Table of Contents

  1. Introduction
  2. A Jump-Growth Model for Size-spectrum in Aquatic Ecosystems
  3. Deterministic Jump-Growth Model with Offspring-production
  4. The Choice of the feeding preference function:
  5. Assumptions on dependencies:
  6. Formal Properties
  7. Moment-Control:
  8. Existence of Solutions
  9. Steady States and Long-time Behaviour
  10. Stationary States
  11. Trivial steady state:
  12. Non-trivial steady state - gaps in the size-spectrum:
  13. Long-time Behaviour
  14. Power-law Equilibria
  15. Numerical Simulations
  16. ...and 7 more sections

Key Result

Lemma 1

Let $r \in [a,b]$ such that $0<a<b$. Then for $K<1$ given and $K'$ chosen accordingly small enough, there exist constants $0<\tilde{m}<1$ and $1<m_*<\infty$ such that we have and

Figures (10)

  • Figure 1: Visualisation of the underlying individual based interaction rules, governing the dynamics encoded by equation \ref{['e:PJG']}. A predator with weight $w$ grows and jumps to state $w+Kw'$, where $w'$ is the prey weight, while producing an amount $\frac{1-K}{K'}$ of 'offspring' of size $K'w'$ as a side-product of the biomass, which could not be assimilated by the predator.
  • Figure 2: Both choices of the feeding preference function $s(\cdot)$ plotted against predator-prey ratio $r$. On the left, the Gaußian feeding preference \ref{['d:sgauss']}. On the right the feeding preference with compact support \ref{['d:scomp']}. The parameters $B=1.5$ and $\sigma=0.3$ were chosen.
  • Figure 3: Function $F$\ref{['d:F']} plotted against moments $m$ with various values $r \in \operatorname{supp}{(s)}$, with $s(\cdot)$ as \ref{['d:scomp']}, $B=1.5$, $\sigma=0.3$, $K=0.3$ and $K'=0.1$.
  • Figure 4: Support of non-trivial steady state represented with solid lines, while the gaps in the size spectrum are given by dashed black lines. For this demonstration the values $B=1.5$, $\sigma=0.3$ and the reference weight $\bar{w}=17$ were chosen.
  • Figure 5: Simulations with compactly supported feeding preference function starting from linear initial conditions with parameters $\alpha=0.9$, $B=1.5$, $\sigma=0.3$, $K=0.1$ and $K'=0.01$, showing convergence to a steady-state representing the cascade-effect.
  • ...and 5 more figures

Theorems & Definitions (12)

  • Remark 1
  • Remark 2
  • Lemma 1
  • proof
  • Remark 3
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Remark 4
  • ...and 2 more