A Stable Finite-Difference Scheme for Population Growth and Diffusion on a Map.
W P Petersen1,2, S Callegari1, G R Lake3
1Anthropological Institute and Museum, Univ. of Zürich, Zürich, Switzerland.
Plos One
|January 14, 2017
Summary
We developed a stable numerical method to simulate human population dispersal using the Fisher-Kolmogorov equation. This approach models population spread over maps, predicting human arrival times in the Americas.
Area of Science:
- Computational modeling
- Population dynamics
- Paleoecology
Background:
- The Fisher-Kolmogorov-Petrovski-Piskunov (FKPP) equation models population growth and diffusion.
- Simulating human dispersal requires stable numerical methods applicable to complex geographical and environmental changes.
Purpose of the Study:
- To present a stable, semi-implicit Godunov-type splitting method for FKPP equation simulations.
- To apply this method to model late Pleistocene human population dispersal across geographical maps.
Main Methods:
- Developed a general Godunov-type splitting procedure for numerical simulations.
- Implemented Neumann boundary conditions for simulations on a world map.
- Utilized Net Primary Productivity (NPP) as a proxy for carrying capacity.
Main Results:
- The semi-implicit procedure ensures high numerical stability.
- The model successfully simulates population dispersal over geographical landscapes.
- NPP data was used to predict human arrival times in the Americas during the late Pleistocene.
Conclusions:
- The developed numerical method is effective for simulating population dispersal.
- This approach provides insights into human migration patterns and timing.
- Paleovegetation and paleoclimate data, proxied by NPP, are crucial for accurate dispersal modeling.
Related Concept Videos
01:25Modeling with Differential Equations
138
Population dynamics can be described mathematically by considering the population size P(t) as a function of time. The rate of change of the population is then represented by the derivative of P(t). A simple assumption is that the rate of growth is proportional to the size of the population itself. This leads to an exponential growth model, where the population increases rapidly without bound. While this is a useful first approximation, it does not reflect realistic long-term...
138
00:57Population Growth
29.3K
Population size is dynamic, increasing with birth rates and immigration, and decreasing with death rates and emigration. In ideal conditions with unlimited resources, populations can increase exponentially, which plots as a J-shaped growth rate curve of population size against time. This type of curve is characteristic of newly-introduced invasive species, or populations that have suffered catastrophic declines and are rebounding.
29.3K
01:27Growth Models with Integration: Problem Solving
84
In population modeling, integration provides a systematic way to determine accumulated quantities from known rates of change. One such application arises in ecology, where the total weight of a fish population in a body of water is referred to as its biomass. When the rate of growth of this biomass is known as a function of time, calculus can be used to determine the total biomass at a future date.Growth Rate and Biomass FunctionLet the growth rate of the fish population be represented by a...
84
02:33Exponential Equations for Modeling Growth
302
Exponential models are essential for describing rapid, multiplicative changes in natural systems, such as population growth. When a population doubles at regular intervals, the process can be modeled using a suitable base. For instance, a bacterial culture that doubles every three hours follows the model n(t)=n0⋅2t/3, where n(t) is the population at the time t.A more general model uses the natural base e, especially for continuous growth. This takes the form n(t)=n0⋅ert, where r is...
302
01:29Exponential Growth
105
Bacterial populations exhibit exponential growth when conditions such as nutrient availability and temperature are favorable. In this phase, cells reproduce through binary fission, where each cell divides into two identical daughter cells. This process causes the population to double at regular intervals, resulting in a growth rate that is directly proportional to the current number of cells. As the population increases, the number of new cells formed during each generation also grows, creating...
105
01:21Diffusion
7.0K
Diffusion is a type of passive transport. In passive transport, a substance tends to move from an area of high concentration to an area of low concentration until the concentration is equal across the space. For example, take the diffusion of substances through the air. When someone opens a perfume bottle in a room filled with people, the perfume is at its highest concentration in the bottle and is at its lowest at the edges of the room. The perfume vapor will diffuse, or spread away, from the...
7.0K