The probability of getting a territory,

*P*_{e}, is given by the territory function and is a function of the time of arrival

where

is the day of arrival corresponding to

*P*_{e}=0.5 and

*z* is a measure of competition for territories such that

*P*_{e} approaches a step function (equal to intense competition) when

*z* goes to infinity. There are situations when there is no single evolutionary stable optimal arrival date (see

Iwasa & Levin 1995) and we should therefore expect a distribution of arrival dates. However, we are studying the optimal arrival of the average individual and our predictions can be confronted with empirical data on the average behaviour of a population. Distinguishing empirically between variance in arrival dates resulting from different optima within a population or different strengths of density-dependent competition across populations is difficult and requires very detailed data (e.g.

Gunnarsson *et al*. 2006). Competition for territories can be seen as a game and there will be a strong selection for arriving early enough to get a territory (

Kokko 1999). To mimic this selection, even though all individuals are assumed equal, we first find the optimal arrival date (

*t*^{*}) for a given

(see below). We then decrease

and find the new

*t*^{*} until

, such that there will not be any benefit arriving earlier than

.

When the bird arrives, it takes

*x*_{1} days before egg-laying can be initialized. One can think of

*x*_{1} as the time needed to find a mate, a territory, accumulate resources and/or recover from migration depending on the life-history characteristics of the species. During the time period

*x*_{1}, there is a risk of mortality. The mortality cost modelled here is the extra cost of early arrival in the breeding area, and the longer the period

*x*_{1} the lower the probability of surviving to breeding. Furthermore, the instantaneous mortality rate goes down later in the season, so that it becomes easier to survive

*x*_{1} days later than early in the season. The probability of surviving from the day of arrival (

*t*_{0}) to the start of breeding (initializing egg-laying),

*t*_{1}=

*t*_{0}+

*x*_{1}, can be expressed as

where

*λ*_{max} is the maximum mortality rate and

*b* measures the strength of the exponential decline rate over the season (see

appendix A for the derivation of equation

(2.2)). Throughout the paper, we will use a fixed value of

*λ*_{max} and vary the mortality risk by changing

*b*. To keep the model simple, we assume that there is no mortality risk during breeding (

Sillet & Holmes 2002). However, the results would not be qualitatively different if we instead made the assumption of a low daily mortality rate throughout the breeding season.

Even though there are obvious fitness gains from arriving early, the onset of breeding has presumably evolved through natural selection to match offspring requirements and food availability (

Lack 1968;

Visser *et al*. 2004). The amount of food available over the season is described by a normal random variate,

*r*_{t}, with mean

*μ* and standard deviation

*σ*. The amount of food available during breeding (

*R*) is the integral of

*r*_{t} between the start of breeding (

*t*_{1}) and the end of breeding (

*t*_{2}=

*t*_{1}+

*x*_{2}),

where

*x*_{2} is the fixed time to accomplish the breeding. We evaluated this integral numerically using the quad function in

MatLab (

The MathWorks 2000). One could, of course, think of various nonlinear relationships between the amount of food available and the number of offspring produced. However, as a first approximation we assume a linear relationship. Hence, the life-history problem is to find the day of arrival maximizing the expected number of offspring, hence, to maximize the product

*P*_{e}*P*_{s}*R*.

(a) Climate change and life history

We consider two alternative climate change scenarios resulting in (i) a shift towards earlier resource peak date or (ii) an increase or decrease in the variance of the resource distribution. We contrasted two different life histories characterized by high/low degree of competition for territories. Furthermore, we studied the effect of pre-breeding mortality risk by varying the seasonal survival curve. For each life history and survival curve, we calculated the optimal arrival date under the climate change scenarios.