01-fundamental-law.Rmd

# The fundamental law of population dynamics {#chap:fundamental}

**Learning objectives**
At the end of chapter 1, the students should be able to: 

General objective: 
1. Differentiate between the different terms and concepts in wildlife management, population models, and the law of population dynamics.

Chapter objectives: 
1. Identify the different components of the fundamental equation. 
2. Understand the difference between objective, subjective, and intersubjective knowledge. 
3. Understand the link between intersubjective knowledge and decision making in wildlife management. 

## Introduction

\begin{equation}
  \Delta N = B - D + I - E
  (\#eq:fundamental)
\end{equation}

This simple equation governs all changes in single species populations. The change ($\Delta$) in the abundance of a
population in a given area across an interval of time, $\Delta N$, is the sum of the number of births, $B$, the
number of deaths, $D$, the number of immigrants, or individuals arriving in the area from outside, $I$, and
the number of emigrants, individuals leaving the area, $E$. As the interval of time gets smaller, we can write the fundamental law as a rate

\begin{equation}
N' = bN - dN + i - e
(\#eq:fundamental2)
\end{equation}

where we’ve replaced the number of births and deaths with the products of the population abundance, N, and a per capita rate of birth and death. We’ve left immigration and emigration as fixed rates. The apostrophe notation for N means "instantaneous rate of change", that is, the rate when the time interval approaches 0, $\Delta \rightarrow 0$.

Where things get interesting is when one or more of the rate constants (B, D, I, E) in \@ref(eq:fundamental2) or amounts (b, d, i, e) in \@ref(eq:fundamental) on the right hand side of these equations are not constant. In particular, when they depend on $N$, or on the abundance of other species, the dynamics of populations get very interesting.

### Killer whales and sea otters

Sea otter (*Enhydra lutris*) populations recovered in Alaska after nearly a century of overhunting (fur trade). Their populations saw a recovery after the International Fur Seal Treaty began in 1911. However, in Western Alaka scientists in the late 20th century noticed their numbers declining again. Researchers radio-tagged sea otters from 1992-1996 to determine birth rates of adult females and pup survival and discovered that these rates were similar to those of stable populations. So if adult females and pups were surviving at the same rate as other populations in which no decline was observed, then what could posibly be causing the declines? 

The answer was increased mortality. The culprit this time was killer whales (*Orcinus orca*). Scientists estimated that a single killer whale could consume 1825 otters per year (assuming this is the only prey they are consuming). However, sea otters and killer whales have co-existed in west-central Aleutian archipielago for thousands of years. Why then this sudden predation effect on sea otter populations? The most likely explanaition is a shift in the prey resource base for killer whales, who are known to prey on marine mammals such as Steller sea lions and harbor seals. The difference now is that the populations of both these pinnipeds have collapsed recently in the western North Pacific. In this example, the population dynamics of sea otters depended on the abundance of Steller sea lions and harbor seals. 

(Ref: Estes et al. 1998)

## The laws of nature

The title of this chapter describes equation \@ref(eq:fundamental) as a "law" -- what do I mean by law? Is it something that was out there, waiting to be discovered by humans, independent of our existence and thought? Or was it created by our thinking of it, consistent with reality but not of reality? Joe Rosen, formerly professor of physics at Tel Aviv University and the University of Central Arkansas, spent an entire volume thinking hard about these issues [@rosen2010lawless]. His categorization of reality and what we can know about it is useful and easy to follow, so I will use it here. He begins with the notion that there is an objective reality that exists independent of our existence. The primary reason for this observation is the simple fact that nature pushes back. Imagine a world where you can fly; wouldn’t it be marvelous! If the world were not objective, but merely a construct of our imagination (a view of reality known as solipsism), then you could create this world, and fly. Unfortunately nature pushes back, and you will fall to the ground. So objective reality constrains what we can do.

The opposite of objective is subjective. Our inner thoughts and feelings are subjective, that is, they are known only to us as individuals. You might tell me what you are thinking or feeling, but I have no independent way of verifying that information. Beliefs about objective reality are similarly subjective, in
that two people can have different beliefs about reality. **However, it is possible for us to conduct *reality checks* on our beliefs about reality. If enough of us get together to check our beliefs, and over time, agree to a consensus belief that passes reality checks, then this is about as close to objective knowledge as we can get. Rosen calls this form of knowledge *intersubjective*;** it is different from subjective belief by virtue of its broader consensus amongst many people, and yet not fully objective by virtue of the fact that it was formed from our subjective perceptions of reality. 

Intersubjective knowledge is socially constructed knowledge, but is not consistent with the post-modernist position that all reality is socially constructed. Our socially constructed, intersubjective beliefs are constrained by objective reality -- not everything is possible. Even if a diehard post-modernist could convince a group of a 1000 people that she could fly unaided, she would not be able to do so. 

**In the field of wildlife management science, the goal is the production of “reliable knowledge” [@romesburg1981wildlife] to use in making management decisions. It is not uncommon to see exhortations from leaders in the field to make science based decisions, presumably a call to use reliable, or intersubjective, knowledge to decide which course of action should be followed.** Unfortunately, as we will see in many examples throughout this book, people do not make decisions like that. Our subjective beliefs about many things, from religion to justice, affect what we think should be done. Inevitably, the more people are affected by a decision or policy related to wildlife management, the more political (i.e. subjective) the decision or policy will become.

So a *law* of population dynamics is well tested intersubjective knowledge, or reliable knowledge, that we can use to make predictions about the consequences of management actions. As you will see below, a law will also have assumptions that must be met in order for it to apply. 

## The nature of models of nature

Almost all of the equations in this book, other than the fundamental equation, are models of nature. That is, they are deliberate simplifications of what is really going on out in nature. If our models were exact replicas of nature, then we would have as much trouble understanding the model as we have understanding nature! It is easy to get caught up in the thrill (OK, easy for people who build models) of adding ever more biological realism to a model. But this does not necessarily help us make decisions in the face of complexity.

A consequence of this deliberate simplification is that all models are in fact, fictitious. This might seem surprising. After all, works of fiction are by definition false. Not true. How can something that is known to be false help us understand nature? This is a problem that so vexed early philosophers of science that they deliberately ignored it for the better part of a century.  Fortunately for the science of population dynamics, this philosophical high-mindedness did not stop scientists from using models to understand nature. Statistician George Box put it this way: 

> All models are wrong. Some models are useful.
> --- @box1976science. 

The key in Science is identifying which models are useful. In this book the utility of a model is measured by the extent to which the model allows us to forecast the future consequences of management decisions.


## Glossary

1. **Solipsism:** A metaphysical viewpoint that asserts reality is only the result of our imaginations.
2. **Model:** a mathematical model is a description of a system using mathematical language and concepts to better understand the effects of different components in the system.