The Mathematics of Antarctic Krill

Keywords: Optimization, chaos theory, krill herd algorithm, Lotka-Volterra, Leslie-Gower

Abstract

This paper explores the mathematical principles underlying the adaptive survival strategies of Antarctic krill, focusing on swarm optimization and predator-prey dynamics. Krill swarms exhibit complex, coordinated movement patterns that can be modeled through chaotic dynamics and optimization theory. Using the Krill Herd algorithm, we demonstrate how krill utilizes attraction, repulsion, and alignment forces to form efficient swarms, optimizing energy use and responsiveness to environmental cues. Additionally, predator-prey models, including Lotka-Volterra and Leslie-Gower equations, illustrate how krill populations sustain themselves within ecosystems despite high predation rates. A key finding is the krills high reproductive rate, which supports rapid recovery from predation events and ensures population resilience. This study provides a mathematical framework for understanding krills evolutionary adaptations, offering insights into natural optimization strategies and ecological stability.

Introduction

Antarctic krill (Euphausia superba) are small but incredibly important creatures in the Southern Ocean. They are a key part of the marine food chain, feeding animals like fish, seals, and whales, and their combined biomass is one of the largest of any species on Earth (Shao et al., 2023). Krill evolved about 130 million years ago during the Lower Cretaceous period and has developed unique adaptations that help them survive in cold, harsh environments (Jarman, 2001). These adaptations, such as forming large swarms and reproducing quickly, allow them to handle threats like heavy predation while maintaining their population. Krill lives mostly in the Southern Ocean, where they gather in huge swarms that make feeding on phytoplankton more efficient. These swarms, which are some of the largest groups of animals in the world, are essential for the health of the Antarctic ecosystem. Krill also plays an important role in cycling nutrients and carbon through the ocean (Cavan et al., 2019). By using mathematical models, we can study how krill use attraction, repulsion, and alignment to move together in efficient swarms. The Krill Herd algorithm, based on their behavior, helps us understand how they optimize movement to find food and avoid predators. Predator-prey models, like Lotka-Volterra and Leslie-Gower equations, show how krill manage to maintain their population even under heavy predation. In this paper, we look at krill behavior from a mathematical perspective to better understand how these tiny animals achieve such efficiency and resilience. This approach demonstrates how math can explain nature and highlights the crucial role krill play in maintaining the stability of the Antarctic ecosystem.

Chaotic Krill Herd Algorithm and Optimization Theory

The process of optimization involves selecting a vector within a search space that maximizes or minimizes an objective function to provide the best solution, a global optima. Typically, nature-inspired intelligent methods are employed to address these types of optimization problems. These optimization theories can be meta-heuristic, meaning they are problem-independent algorithms that make a trade-off between intensification (local search) and randomization (global search). The basic Krill Herd (KH) algorithm is a swarm intelligence meta-heuristic optimization method that mimics the herding behavior of krill (Wang et al., 2014). The objective function is the distance of food location and the position of the krill (Wang et al., 2017). In the KH algorithm, individual krill position is influenced by three primary components:

1. Movement induced by other krill: This allows krill to adjust their positions based on the behavior of their peers, enhancing group cohesion and safety from predators.

2. Foraging action: Each krill's movement is influenced by the location of food and their previous experiences with food sources, which is crucial for survival and energy acquisition.

3. Random diffusion: This introduces a level of unpredictability in movement, which can help krill evade predators by making their movements less predictable.

The movement of each krill can be simplified to the following Lagrangian model (Eq. 1):

Where N represents the induced movement, Fi is the foraging motion, and Di denotes the random diffusion for the krill. N is calculated based on three factors: local effect, target effect, and repulsive effect. Local effect refers to the influence that nearby krill have on an individual krill's movement and is determined by a sensing distance as seen in Figure 1.

Fig. 1. A schematic representation of the sensing radius around an individual krill (Gandomi, 2012).

Target effect is based on the krill's desire to move towards a specific target, such as a food source. The repulsive effect acts as a counterforce that prevents krill from clustering too closely together. This is particularly important as krill need to maintain personal space to evade predators. F is determined by food location and previous experience regarding food. Food location refers to the specific areas in the environment where food is available. Krills are motivated to move towards these food sources to maximize their energy intake. Previous experience encompasses the krills past interactions with food sources. If a krill has successfully foraged in a particular area before, it is likely to return to that location based on its memory of food availability. This experiential learning allows krill to make informed decisions about where to forage, enhancing their chances of finding food in the future. D is a random process that is calculated from the maximum diffusion speed and a random vector. Maximum diffusion speed defines the upper limit of how fast a krill can diffuse through its environment. It is crucial for ensuring that the krill can explore their surroundings effectively while avoiding predation. The random vector is a variable that introduces randomness into the krill movement. It is defined within the range of [−1,1], allowing for a diverse set of movement directions.

Though the KH algorithm is highly effective, it can struggle with global search, leading to occasional failures in finding the global optimal solution. Its search strategy primarily relies on random walks, which can sometimes be inadequate for successfully exploring the entire search space. Inertia weights help control the balance between exploration (searching new areas) and exploitation (refining known good solutions). A higher inertia weight encourages exploration, while a lower inertia weight promotes exploitation. In the standard KH algorithm, inertia weights are set to 0.9 at the early search stage to emphasize exploration. As the algorithm progresses, these weights are linearly decreased to 0.1. This gradual reduction stimulates exploitation, allowing krill to focus on refining their positions around promising food areas in the search space. However, this linear decrease can cause the algorithm to get stuck at a local optimum (and not find the global optimum) because it is not exploring sufficiently. This is where chaos theory comes in.

Chaos theory is the study of apparently random or unpredictable behaviors in systems governed by deterministic laws and can be applied to the KH algorithm to create the Chaotic Krill Herd (CKH) algorithm. This incorporation of chaos allows for more efficient exploration of the search space, leading to faster and more reliable convergence of optimal solutions. To enhance the performance of the KH algorithm, twelve chaotic maps are utilized to adjust the inertia weights (movements of krill individuals) dynamically (Wang, 2014). Chaotic maps are mathematical functions used to generate sequences that exhibit chaotic behavior, meaning they are highly sensitive to initial conditions and appear random. A key property of chaotic systems is their sensitivity to initial conditions, often known as the “butterfly effect”, where small changes in starting points lead to vastly different outcomes. In the context of optimization, chaos introduces a controlled, complex variability to parameters. This variability enables the algorithm to jump out of local optima (regions where it might get “stuck”) by constantly adjusting in unpredictable ways.

The video below shows a general Herding Algorithm in action (Fig. 2). It is not the CKH algorithm, but for lack of CKH videos, it still displays how algorithms can model real life behavior. The following video demonstrates an algorithm that solves the “shepherding problem.” It is based on adaptive switching between collecting the agents when they are too dispersed and driving them once they are aggregated, similarly to how krill switch between exploration and exploitation.

Fig. 2. Video displaying how herding algorithms can model real life behavior and solve problems (Strömbom, 2014).

By simulating chaotic behavior, the CKH algorithm can better balance exploration (searching across broad areas) and exploitation (focusing on the most promising regions), transforming the optimization process into one that is more dynamic and adaptable. So, while krill agents update their positions based off the factors; neighboring influence, food attraction and diffusion, chaos theory introduces irregular changes into these movements, so that the krill can avoid converging prematurely. Through the integration of krill-inspired foraging with mathematical chaotic dynamics, the CKH algorithm results in a powerful, hybrid optimization technique well-suited to complex, high-dimensional search spaces that the krill inhabits.

Predator-prey Dynamics and Trophic Modeling

Blue Whale and Krill Populations Modelling (Lotka-Volterra)

Krill’s population dynamics impact the entire food web, especially for species that rely on krill as a primary food source, such as the blue whale (De Felice et al., 2024). Predator-prey models simulate how these populations respond to and are affected by predation pressures. By using a modified Lotka-Volterra model (Zhang, 2024), it can be demonstrated how krill's high reproductive rate allows them to recover quickly even after significant predation events. The model captures the cyclic nature of predator-prey dynamics, where krill’s high growth rate ensures their population rebounds even when their numbers are reduced. This rapid reproduction is an evolutionary adaptation to ensure their survival despite being a primary food source for many marine predators.

Before setting up the Lotka-Volterra model, the following assumptions are made:

1. B(t) represents the blue whale population in thousands and K(t) represents the krill population density in tons per acre, reflecting their small size and schooling behavior (Zhang, 2024).

2. In the absence of krill, the blue whale population declines at a rate of 80% annually, considering the possibility of alternative food sources (Zhang, 2024).

3. The krill population grows at a rate of 100% annually in a predator-free environment (a = 1) (Zhang, 2024).

4. The blue whale population growth is proportional to the product B(t)K(t), with a proportionality constant of n = 1, based on estimates showing an 8.2% annual growth rate (Zhang, 2024).

5. The krill population declines at a rate proportional to B(t)K(t), determined by the constant b = 0.02 (Zhang, 2024).

6. The initial blue whale population is set at B(0) = 20, reflecting the current estimate of 10,000 to 25,000 individuals (WWF-Australia, n.d.).

7. The initial krill population density is set at K(0) = 5, based on approximations of biomass in the Southern Ocean (Zhang, 2024).

These assumptions form the foundation of the Lotka-Volterra model by defining the populations, growth rates, and interactions between blue whales and krill. The representation of whales in thousands and krill by density simplifies calculations while staying consistent with ecological data. The assumed 80% annual decline in whale populations without krill accounts for their dependency on food as well as potential alternative sources. Whale and krill interactions are captured by proportional growth and decline rates, with constants a and b reflecting the strength of these effects. Finally, initial population values reflect realistic estimates from conservation organizations and ecological studies, ensuring the model starts with credible values. These assumptions provide a simplified yet effective framework for analyzing predator-prey dynamics.

Consequently, the overall rate of change of the blue whale population (dB/dt) is represented by the addition of the growth of the whale population due to krill availability minus the natural decay of the whale population in the absence of krill. Thus, the following differential equation is obtained (Eq. 2):

Regarding the overall rate of change of the krill population (dK/dt) the same thing applies. The decline of the krill population due to predation by blue whales is subtracted from their natural growth in the absence of predators (Eq.3).

Lastly, to set up the initial value problem, the initial conditions are given as B(0) = 20 and K(0) = 5. The implicit solution to this differential equation is given by Eq. 4 where the constant 29.47 approximates (20*5^0.8) / e^0.9.

The Lotka-Volterra model can be visualized through a phase plane (Fig. 3) which forms a closed curve that represents the periodic interaction between prey and predator populations. The model suggests that, while the whale population fluctuates based on krill availability, the krill population will always try to regenerate according to its high reproductive capacity. Even after heavy predation, krill can replenish their numbers relatively quickly, as seen by their high growth rates (a=1).

Fig. 3. Phase plane of the Lotka-Volterra model shows the periodic interaction between prey (krill) and predator (whale) populations. The closed curves represent different population cycles, with the equilibrium point marked. The model illustrates how the krill population can quickly recover due to high growth rates (a = 1) even after significant predation by whales (Zhang, 2024).

There are several equilibrium points (where the population sizes do not change because they are stable) to consider and these can be calculated by setting the differential equation to zero. In Eq. 5, we get B=0 or K=8, while in Eq. 6, B=50 or K=0.

Analysis of the results reveals both trivial and nontrivial equilibrium points. The trivial equilibrium occurs in the absence of prey and predators, corresponding to B = 0 and K = 0. However, the most biologically significant equilibrium is the nontrivial state, where both species coexist in balance. In this stable condition, the blue whale population stabilizes at B = 50 (50,000 individuals), and the krill biomass reaches K = 8 (8 tons per acre).

To demonstrate the high reproductivity of krill, we introduce a disturbance (harvesting, environmental change, etc.) that decreases the krill population by half from equilibrium (K₀ = 4), while the whale population stays constant (B₀ = 50). Euler's method can then be used to solve the system of differential equations numerically and simulate the recovery of krill over time (Eq. 7).

The time step Δt = 0.1 years will be used (Eq. 8 & Eq. 9). As a result, at Δt = 0.1:

As seen above, the population of whales decreased while that of krill stayed the same. If this process is repeated until t =1, K₁ = 4.59 and B₁ = 32.53. Three interpretations can be taken from these values and visualized in Figure 4:

1. The krill population remains stable at around 4 tons/acre, with only small fluctuations due to the whale population's predation pressure.

2. The whale population decreases over time due to the reduced availability of krill as a food source, demonstrating the predator-prey dynamic.

3. Although the krill population is initially relatively low, it maintains its growth potential as the number of whales decreases, thus demonstrating the high reproductivity of the former population.

Fig. 4. Time evolution and phase plane of krill and whale populations. The left plot shows the time evolution of krill (in tons per acre) and whale populations (in thousands) over several years, illustrating the predator-prey interaction. The right plot displays the phase plane of krill vs. whale populations, showing a closed orbit that represents the dynamic equilibrium between the two populations. These visualizations highlight the resilience of krill populations due to high growth rates, allowing them to sustain numbers despite whale predation.

Nonetheless, the model has several limitations which need to be considered. For instance, the model focuses only on the predator-prey relationship between blue whales and krill, ignoring other environmental factors and interactions with other species. This simplification makes the model easier to work with but leaves out important details like changes in ocean temperature, competition for food, and interactions with other marine animals. These factors could have a big impact on whale and krill populations. In addition, the assumptions made—such as growth and decline rates and starting population sizes—are estimates based on current data and do not fully reflect real-world conditions. Adding these missing elements and improving the assumptions could make the model more accurate and useful in the future.

In summary, the Lotka-Volterra model reveals the resilience of krill populations, demonstrating their ability to sustain themselves and support predator populations like the blue whale, despite fluctuating environmental pressures.

Blue Whale and Krill Populations Modelling (Leslie-Gower)

As previously mentioned, the Lotka-Volterra model relies on several assumptions that may not accurately reflect the complexities of real-world ecosystems. For example, there is no carrying capacity for the predator. To address these limitations, the Leslie-Gower predator-prey model provides an alternate approach, where the predator's carrying capacity is proportional to the prey population, highlighting that both prey (x) and predator (y) populations have upper limits to their growth rates (González-Olivares et al., 2012).

The Leslie-Gower model can be further enhanced to be more realistic when combined with Holling-Type II functional responses, which account for the predator’s diminishing prey consumption rate as prey density increases. This saturation effect is represented by Eq. 10.

Where

F(x): Rate of prey consumption
e: Efficiency of the predator in capturing prey

h: Handling time per prey item

x: Prey population density (krill)

This equation shows that at low prey densities; consumption increases almost linearly with prey availability. However, as prey density grows, the handling time limits consumption rates, causing the rate of prey consumption to plateau. This reflects the real-world scenario where an abundance of prey does not lead to a proportional increase in predation due to physical and behavioral limitations in predators (Yue, 2016)

In Antarctic ecosystems, krill – one of the most abundant species - benefit from this dynamic due to their high reproductive rate, allowing their population to sustain itself even under intense predation pressure from whales, as well as seals, penguins, and fish (Shao et al., 2023). This reproductive capacity, combined with their swarming behavior allows krill to form natural “prey refuges”, where swarms protect individual krill by reducing the chance of any one individual being targeted. A portion of the krill population can escape predation, supporting rapid recovery and population stability.

The concept of prey refuges, where a fraction of the prey population remains inaccessible to predators, can be expressed mathematically by Eq. 11.

Where:

P_effective: effective prey population exposed to predation

R: population of prey in the refuge

By limiting the accessible prey population, refuges stabilize predator-prey interactions and prevent over-predation, ensuring ecosystem resilience. This evolutionary strategy allows krill to not only promote their own survival but also act as a keystone species in the Antarctic. Without krill, the Antarctic ecosystem would be severely disrupted, affecting the survival of numerous species that rely on them as primary food source (Cavan et al., 2019).

Considering both the Holling-Type II functional responses, which account for the predator’s diminishing prey consumption rate as prey density increases, and the concept of prey refuges, the predator-prey dynamics can be incorporated into the Leslie-Gower framework by Eq. 12 & Eq. 13.

Where Intrinsic growth rate of the prey population, is carrying capacity of the prey (krill), is Natural mortality rate of the predator, and is conversion efficiency of prey consumption into predator reproduction.

The modification of the Leslie-Gower framework demonstrates the unique ecological position of krill as an abundant and resilient species that supports a diverse range of predators, beyond just whales, the survival of species such as seals, penguins and many fish that depend on the krill population. The stability of the Krill population, with a high growth rate, allows them to replenish their population efficiently. Their reproductive capacity allows krill populations to maintain stable even under intense predation pressure.

Additionally, the swarming effect that limits predation rate as krill population increases creates a natural upper limit on predator consumption. This built-in buffer ensures that krill population persists, supporting the species that depend on them while preventing ecosystem collapse due to over-predation. The Leslie-Gower model captures the resilience of krill population even as they experience fluctuating predation levels, showcasing how krill population sustain themselves through natural cycles of growth and predation.

Conclusion

Two substantial challenges that krill face in their environment are the need to move efficiently in swarms to find food and the ability to recover quickly from heavy predation. This paper explains how krill overcomes these problems using clever natural strategies that can be understood through mathematical models.

One big problem for krill is that swarm movement can waste energy if it is not well-coordinated. To solve this, krill use chaotic movement patterns, which are explained in the Chaotic Krill Herd Algorithm. This algorithm shows how krill adjust their positions using three key forces: attraction toward food, repulsion to avoid crowding, and alignment to move together in the same direction. These forces help krill move efficiently as a group, saving energy and improving their chances of finding food. The addition of chaos theory makes the model more realistic, reflecting how krill can thrive in unpredictable environments by exploring new areas more effectively and reducing inefficient movement.

Another major challenge for krill is dealing with predators like blue whales, seals, and penguins, which can drastically reduce their population. Predator-prey models, such as Lotka-Volterra and Leslie-Gower equations, simulate how krill overcome this through their ability to reproduce rapidly. For example, even if half the krill population is lost, their high growth rate (up to 100% in predator-free conditions) allows them to recover quickly. Additionally, krill swarms create a kind of “safe zone” for some members of the population, reducing the risk of individual predation. This protection, combined with their rapid reproduction, ensures that krill populations can bounce back even after significant losses, keeping the food chain stable.

Krill tackle their challenges with efficiency and resilience, as illustrated through mathematical modelling. Their swarm movements save energy and make foraging more effective, while their reproductive strategies and group behaviours protect their populations from predators. This shows the clever ways krill adapt to their environment, offering insights into how nature develops solutions to complex problems.