= Solution
The <Poisson generalized linear mixed model> assumes conditionally <Poisson distributed> counts with a <log link>. The fitted <variance components> are both zero, so this fit lies on a boundary and reduces to ordinary <Poisson regression>. It estimates
$$
\widehat\mu(a)=\exp(0.793+0.064a),\qquad
\boxed{\frac{\widehat\mu(a+1)}{\widehat\mu(a)}=e^{0.064}\simeq1.066}.
$$
Thus the fitted annual increase is about \b[6.6%], with approximate 95% <confidence interval> for the <rate ratio> $(e^{0.001},e^{0.128})\simeq(1.001,1.137)$. The age evidence is much weaker than in the Gaussian fit: the quoted <p-value> is $0.047$. Zero estimated group variances do not prove that real bird and year differences are absent.
The <Poisson distribution> forces conditional <variance> to equal the conditional <expected value>. Here fitted means are around two to three hatchlings, whereas the Gaussian model estimated a residual <variance> of only $0.188$. This suggests checking <underdispersion> and the bounded nature of hatching counts, rather than assuming a Poisson response solves the diagnostic problems. Inspect simulated count residuals and predictive distributions; if the number of eggs is observed, fit a binomial mixed model and check that model's dispersion too.
The displayed Gaussian and Poisson <Akaike information criterion> values should not simply be ranked as if they used the same likelihood convention: one evaluates a continuous density and the other a probability mass. Compare properly specified count models on the same observed response scale, with compatible likelihoods and predictive checks. A boundary fit, doubtful dispersion assumptions, and possible nonlinear age effects all warrant further work before a strong biological conclusion.
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