March 11, 2017 by Species Ecology
How to become a 21st Century Modern Ecologist: Pathways and Ways Forward
Mathematics: The Common Language Across the Science Sphere
Mathematical biology is growing rapidly. Mathematics has long played a dominant role in our understanding of physics, chemistry, and other physical sciences. However, wholesale application of mathematical methods in the life sciences is relatively recent. Now questions about infectious diseases, cell movement, ecology, environmental changes, and genomics are being tackled and analyzed using mathematical and computational methods. While the application of quantitative analysis in the life sciences has borne fruit in the research arena, only recently has it impacted undergraduate education. Until a few years ago, the number of undergraduate texts in mathematical biology could be counted on one hand. Now this has changed dramatically. Recent undergraduate texts range from simple introductions to biological numeracy, freshman calculus for students in the life sciences, modeling with differential equations, computer algebra, and dynamical computer-based systems to name but a few. Despite the plentitude of new books, mathematical biology is still rarely offered as an undergraduate course. This article is to introduce readers to the mathematical underpinning of life science and our targeted audiences in this remit are the undergraduate or graduate students of developing nations undertaking ecology or wildlife biology as their undergraduate degree. Graduate students of freshman or sophomore year taking up three credits course in numerical ecology will also find it useful as an introductory chapter-reading.
Mathematical Ecology: The Key to become a 21st Century Modern Biologist
Mathematical ecology dwells at the interface of two fields: applied/computational mathematics and ecology. Individually, these fields are growing quickly due to rapidly changing technology and newly emerging sub-disciplines such as wildlife science, conservation biology, agricultural science, water resource management, silviculture, entomology, climate change and so forth. Coupled together, the fields provide the basis for the emerging scientific discipline of mathematical ecology, the focus of which is interdisciplinary scientific problems in quantitative life sciences. What can ecology offer mathematics and computation? Ecological models offer a seemingly endless supply of challenging and interesting nonlinear problems to solve. These nonlinear problems can provide a testing ground for applied mathematical and computational methods, and generate the impetus to develop new mathematical and computational methods and approaches. What can mathematics and computation offer biology? Mathematics and computation can help solve a growing problem in biological research. Data collection, varying from gene sequencing to remote sensing via satellites, is now inundating mathematicians and ecologists with complex patterns of observations. The ability to collect new data outstrips our ability to heuristically reason mechanisms of cause and effect in complex systems. It is the analysis of mathematical models that allows us to formalize the cause and effect process and tie it to the biological observations.
Ecological Models: The Mandatory Course for Conservation Biologists and Resource Managers
Models are tools that represent essential features of a system so that relationships can be analyzed within established boundary conditions. Modeling may be used to simulate natural conditions and scenarios of resource use. Analyses of models can be used to examine potential impacts of a decision. Ecological models are a tool for environmental managers to enhance understanding of both the complexities and the uniqueness of a given situation and its response to management or change. Models allow managers to summarize information on the environment, determine where gaps exist, extrapolate across the gaps, and simulate various scenarios to evaluate outcomes of environmental management decisions.
Types of Models
There are at least three types of models: heuristic, physical, and mathematical. Heuristic models tend to be relatively simple but capture key relationships of the system. They can be depicted as pictures, diagrams, words, or mathematical relationships. Sometimes scientists call these “back of the envelope” models because they can be explained in a small amount of space. Such models are appealing in the sense that they are relatively easy to understand. However, their simplicity may mean that some of the important interactions in the system are not fully characterized. Physical models are scaled-down versions of the real world, typically constructed in three dimensions, and are sometimes used to show changes over time (the fourth dimension). Examples are microcosms, wind tunnels (used to examine aerodynamic properties of airplanes, cars, and seeds), and aquariums (used in studies of fish population dynamics). One interesting example is the use of model streams built of fiberglass in which certain chemicals can be added or the size, shape, and density of substrate materials controlled. Stream water is circulated in these models, and the growth or behavior of fish or invertebrates is observed. The model streams are designed for monitoring their interior at a height of 1 m rather than ground level of natural streams, which makes the experiments easier to observe. Mathematical models describe relationships via numerical formulations. The chosen equations should appropriately reflect the constraints of the question at hand. The assumptions, form, and outcomes of the model need to be realistic for the situation and clearly communicated to the user.
Few Ground Rules Students Must Remember:
1. The model must make common sense. For example, a Leslie matrix model (Leslie 1945) is commonly used to analyze population dynamics but can project infinite growth. To avoid this unbelievable possibility being discussed in the courtroom, Swartzman (1996) introduced a density-dependent fecundity term into the model.
2. A model must be simple enough for the judges, lawyers, and jury members to understand.
3. Jargon must be avoided.
4. The model and its projections must be clearly described; simple illustrative graphics are helpful.
These lessons are general enough to be applicable to mathematical models that might be applied to environmental decisions. The act of modeling is often called an art because there are many ways to express served relationships using mathematics, and it takes experience, expertise, and creativity to appropriately capture complex interactions. Because of the wider use and range of applicability of mathematical models, they are the focus of ecological and life science.
The Historical Sketch of Ecological Modeling
The roots of ecological modeling for environmental management lie in attempts to explain human population dynamics. The earliest explorations of geometric progression provide a way to explain human population growth (Hutchinson 1978). Recognition that an exponential growth of people up to the sixteenth century would lead to an unrealistic estimate of the people on the Earth in the future required rethinking the use of an uncontrolled exponential growth curve. Verhulst (1838) discovered that the leveling of population growth could be represented by “the logistic equation.” The term “logistic” has a rare meaning of “calculation by arithmetic,” which may explain the use of the term. Data from animal populations showed that exponential growth was not often observed but that an “S” curve of population growth was more typical (i.e., it could be calculated from the data). But the logistic equation was not adopted in the analysis of population growth until studies with laboratory animals confirmed that a saturation point was typically attained. Lotka (1924) expanded upon Verhulst’s work with the logistic equation to come up with the formula that is still in use today. Volterra (1926) expanded the use of the logistic equation to describe the populations of competing species and developed the first published case of an ecological model being used for resource management. The results of his model were applied to explain changes in the proportion of fish in the Mediterranean Sea that resulted from the suspension of commercial fisheries during the war years of 1915 to 1918. Gause subsequently (1934) provided experimental confirmation of these interactions. A decade later, Nicholson and Bailey (1935) used finite difference models to examine parasitism and predation (critical agricultural problems). Difference models rely on discrete time steps rather than the continuous time steps of differential equations. Therefore, difference equations are closer to the data collected at regular intervals by biologists measuring population changes. However, the mathematical properties of differential equations are more easily solved by analytical techniques so they quickly become more widely accepted. Today, both types of approaches can be implemented in computer models. Building upon earlier applications, Hutchinson (1954) constructed mathematical models of population regulation to argue for the importance of feedback loops, which are integral to resource management. His insistence on a rigorous approach to ecology led several of his students to invoke mathematical techniques. Robert MacArthur added a quantitative analysis to the field of community ecology in the development of the concept of competitive exclusion (MacArthur 1958), which led him to the hypothesis that competition determines relationships of species occupying the same area (MacArthur 1960). At about the same time Leslie (1945, 1948) developed a matrix approach to examine changes in life stages over time, and that technique eventually became a common tool in resource management. While working at the Bureau of Animal Population at Oxford, Leslie used matrix algebra to express age-specific relationships (Leslie 1945), explore logistic population growth and predator–prey relations (Leslie 1948), and consider time lags (Leslie 1959). Lefkovitch (1965) built upon Leslie’s ideas but classified individuals by development stage rather than age. This stage approach was also used by Usher (1966) to classify trees. However, these matrix approaches were not adopted by the broad ecological community for about 25 years. Part of the lag in the application of these ideas was the large amount of computation required.
Tips for Students: Linux is the Ultimate Frontier to Learn Mathematical Programming and Ecological Modeling
The development and application of ecological models are tied to the development of computers. Computer availability and flexibility enhanced the usability of models. For example, Caswell (2001) notes that the lack of computational speed hampered the adoption of the matrix models introduced by Bernardelli (1941), Lewis (1942), and Leslie (1945). Instead of those approaches, life-table methods developed at about the same time (Birch 1948; Leslie and Park 1949) were more accessible and could perform most analyses offered by matrix models without the use of computers. The history behind the development of computers goes several centuries back. The abacus is an ancient manual arithmetic device first used by the Chinese to add, subtract, multiply, and divide and to calculate square roots and cube roots. It consists of a frame with moveable counters. The first mechanical calculating machine was invented in the 1600s. During the 1830s, the English mathematician Charles Babbage developed the idea of a mechanical digital computer, but the existing technology was not advanced enough to provide the precision parts needed, and Babbage was not able to secure funding to develop the device. In 1930, Vannevar Bush, an American electrical engineer, built the first reliable analog computer. Many improvements were made during the next decades, but it was John Van Neumann’s idea that programs could be coded as numbers and stored in a computer’s memory that hailed the next major advance. This idea was used in developing the first stored-program digital computer built in 1949. The invention of the transistor in 1947 and related solid-state devices in the 1950s and 1960s helped produce faster and more reliable computers. The first computers represented numerical data by analogous physical magnitudes or electrical signals, whereas later models used binary digits. The move from analog to digital computers increased the speed of computations. Subsequent miniaturization of computers was based on electronic advances in the 1960s and 1970s and led to the wider dissemination of computers. The development of personal computers, the Internet, and mass-storage devices led to a proliferation of hardware and software capabilities that are now basic to many ecological models used for resource management. Today, computers are available and related to almost all aspects of business, communication, and education. In fact, children’s games examine population dynamics in a mathematically sophisticated manner. A continuing challenge, however, is to include the most up-to-date ecological understanding and required complexity in models and to get those models into the hands of resource managers.