Size- and Age-Dependent Natural Mortality in Fish Populations: Detailed Study Notes

Author Information

Kai Lorenzen
School of Forest, Fisheries, and Geomatics Sciences, University of Florida, Gainesville, FL 32653, USA
Handled by: Jim Ianelli

Keywords

Mortality
Size-dependence
Age-dependence
Senescence
Density-dependence
Intrinsic mortality
Fisheries stock assessment
Fisheries management

Abstract

Natural mortality rates (M) in fish populations vary significantly depending on body size and age. Traditionally, fisheries models have treated M as a constant in the recruited stock. However, this assumption is being challenged both theoretically and empirically. This review synthesizes biological, empirical, and theoretical models of size- and age-dependent natural mortality giving rise to a new paradigm called ‘generalized length-inverse mortality’ (GLIM). This paradigm posits that mortality declines inversely with body length throughout much of the juvenile and adult fish lifecycle. The article outlines the implications of mis-specifying size- and age-dependent M in stock assessments, which can range from moderate to severe depending on the modeling context.

1. Introduction

Natural mortality is crucial in fish population dynamics but remains challenging to quantify. Historical models, such as Beverton and Holt’s yield-per-recruit, have relied on a simplified life cycle representation in which natural mortality is treated as a constant rate. Increasingly, evidence and theoretical frameworks have demonstrated size- and age-dependent mortality. Factors driving this shift include:

Evidence showing variation in mortality rates (Vetter, 1988; McGurk, 1986).
Changing fishing pressures affecting juvenile populations (Brodziak et al., 2011).
The emergence of marine reserves leading to the presence of older fish (Berkeley et al., 2004).

2. Biology of Natural Mortality in Fishes

2.1. Conceptual Framework

Natural mortality rates in fish often exhibit a 'bathtub' pattern, showing a decline early in life and a rise later, identified through empirical studies on zebrafish (Danio rerio). Mortality rates are influenced by:

Extrinsic factors: Environmental conditions, predation pressure, competition, and disease.
Intrinsic factors: Senescence, reproductive costs, and physiological resilience.

2.2. Biology and Ecology of Mortality in Fish Populations

Mortality rates decline with size and age due to both extrinsic and intrinsic mechanisms. Predation, for example, is size-dependent, with larger fish predating smaller ones due to size and gape limitations, which further enhances survival rates of larger individuals. As fish populations grow, their ability to escape from or avoid predators increases, thereby influencing their overall mortality patterns.

2.3. Insights from Captive Fish Populations

Captive fish populations, such as zebrafish studied under controlled conditions, reveal intrinsic patterns of mortality. Their mortality rates show a distinct declining pattern with age and size, yet these may not fully represent wild populations due to domestication effects (Thorpe, 2004).

2.4. Size- or Age-Dependent?

Mortality patterns are inherently linked to both size and age, with evidence suggesting that size-based processes predominantly govern mortality relationships in fish populations. The relationship between mortality and length can often be represented mathematically.

3. Mortality Models and Quantitative Generalizations

3.1. Size-Based Allometric Scaling Models

Models for mortality based on size (length L or weight W) take the form:

$M(L)=M\left(Lr\right)\left(\frac{L}{Lr}\right)^{c}$
$M(W)=M\left(Wr)(\frac{W}{Wr}\right)^{b}$

These models have shown mortality scaling as length-inverse particularly at the population level (c ≈ -1). Where MLr and MWr are the mortality rates at reference length Lr or weight Wr

3.2. Age-Based Models

Age-based models derive from size-based models and typically utilize the von Bertalanffy growth function. Some key models include:

Lorenzen M: $M(a) = ML∞ imes (1-e^{-K(a-a_0)})^c$
Charnov M: $M(a) = K imes (1-e^{-K(a-a_0)})^{-1.5}$

3.3. Multi-Species or Ecosystem Models of Predation Mortality

These models separate natural mortality into predation (M2) and residual (M1) mortality components, often needing calibration against empirical data (e.g., dietary data of predators).

3.4. Modeling Density-Dependence in Early Juvenile Mortality

Density-dependence in fish populations primarily affects juveniles, and several approaches exist for modeling these dynamics based on two-stage models of stock recruitment.

4. Synthesis

4.1. Emerging Paradigm: Generalized Length-Inverse Mortality (GLIM)

The GLIM concept establishes a coherent framework where mortality can be modeled as inversely related to length. This not only captures the average patterns observed but also allows for modifications to consider senescence and density effects in juvenile stages.

4.2. Operationalizing GLIM

GLIM can be applied in both length-based and age-based models, facilitating broader use in stock assessments and enabling better representation of intrinsic mortality factors.

4.3. Implications of GLIM for the Theory of Fishing and Related Applications

Adoption of GLIM suggests a paradigm shift in understanding natural mortality, emphasizing higher rates of natural mortality especially in juvenile fish stocks when compared to constant M assessments. This indicates the necessity for better management practices to protect smaller cohorts during harvesting.

5. Consequences of Mis-Specifying the Functional Form of M in Stock Assessments

Mis-specification can lead to significant biases in estimates of stock status and impacts of management strategies, emphasizing the need for rigorous modeling of natural mortality to ensure sustainable fisheries.

6. Conclusions

The GLIM framework synthesizes biological understanding and empirical models, providing a robust modeling approach that should replace the traditional constant M assumption in fisheries assessments. Continued research is necessary to improve predictions and address nuances such as age and density-dependence in mortality dynamics.

1. Introduction to Natural Mortality (M)

Traditional View: Treated as a constant rate in adult fish stocks.
Emerging View: Variable based on size and age, leading to the GLIM (Generalized Length-Inverse Mortality) paradigm.
Drivers of Change: Improved empirical evidence, changing fishing pressures on juveniles, and observations from marine reserves.

2. Biological Mechanisms

Mortality Pattern: Often follows a 'bathtub' curve (high early in life, low in mid-life, high during senescence).
Key Factors:
- Extrinsic: Predation (linked to gape limitations), environment, and disease.
- Intrinsic: Senescence (aging) and reproductive costs.
Size vs. Age: Size is generally the primary driver of mortality relationships in wild populations.

3. Quantitative Mortality Models

Size-Based Scaling: Mortality typically scales inversely with length ( $c \approx -1$ ).
- $M(L)=M(L)=MLr(L/Lr)^{c}$
- $M(W)=M{Wr}(W/W{r})^{b}$
Age-Based Models: Derived using growth functions (e.g., von Bertalanffy).
- Lorenzen M: $M(a)=ML\infty\cdot(1-e^{-K(a-a_0)})^{c}$
- Charnov M: $M(a)=K\cdot(1-e^{-K(a-a_0)})^{-1.5}$
Ecosystem Models: Differentiate between predation mortality ( $M2$ ) and residual mortality ( $M1$ ).

4. The GLIM Paradigm

Concept: Mortality is inversely related to body length throughout most of the life cycle.
Operational Use: Integrated into both length-based and age-based stock assessments.
Implications: Predicts higher mortality in juveniles; necessitates better protection for smaller cohorts during harvesting.

5. Management Consequences

Mis-specification Risks: Assuming constant $M$ when it is size-dependent leads to biased assessments of stock health.
Recommendation: Replace the 'constant $M$ ' assumption with the GLIM framework to ensure sustainable fisheries management and more accurate stock status estimates.