I'm looking for the solution of partial differential equation $$\frac{\partial}{\partial t} f(i,t) =\left(a f(i,t) + b\int_0^\infty \mathrm {d} i\, h(i) f(i,t)\right)h(i)$$
Where $$f(i, 0)=1, \int_0^\infty\mathrm {d} i\, h(i)=1$$
Interesting special case: $h(i)=(i+1)^{-2}$, $a=-2$, $b=1$, want to know $g(t)=\int_0^\infty \mathrm{d}i\ h(i)f(i,t)$.
Background: $g(t)$ gives effect on environment of bird population at time $t$ under continuous version of bird population dynamics model described here
The Laplace transform of the function in question can be calculated rather easily. One can Laplace transform in the time coordinate to obtain
$$s\hat F(i,s)=f(i,0)+ah(i)\hat F(i,s)+bh(i)\int_0^\infty h(x)\hat F(x,s)dx$$
Solving for the $\hat F$ yields $$\hat F(i,s)=\frac{f(i,0)}{s-ah(i)}+\frac{bh(i)}{s-ah(i)}\int_0^\infty h(x)\hat F(x,s)dx$$ Integrate the equation against $h(i)$ and solve the now self-consistent equation to obtain the value of the inner product $\langle h,\hat F\rangle:=\int_0^\infty dx h(x)\hat F(x,s)$
$$\langle h,\hat F\rangle(s)=\frac{\int_0^\infty dx\frac{f(x,0)h(x)}{s-ah(x)}}{1-b\int_0^\infty dx\frac{h^2(x)}{s-ah(x)}}$$
For the specific values provided in the question, the integrals can be computed analytically:
$$\int_0^\infty \frac{h(x)}{s+2h(x)}dx=\frac{1}{\sqrt{2s}}\tan^{-1}\sqrt{\frac{2}{s}}$$
$$\int_0^\infty \frac{h^2(x)}{s+2h(x)}dx=\frac{1}{2}\left(1- \sqrt{\frac{s}{2}}\tan^{-1}\sqrt{\frac{2}{s}}\right)$$
which finally yields for the two functions in question
$$\langle{h,\hat F}\rangle=\frac{\tan^{-1}\sqrt{\frac{2}{s}}}{\sqrt{\frac{s}{2}}\left(1+\sqrt{\frac{s}{2}}\tan^{-1}\sqrt{\frac{2}{s}}\right)}$$
$$\hat F(i,s)=\frac{(i+1)^2}{2+s(i+1)^2}+\frac{1}{s(i+1)^2+2}\frac{\tan^{-1}\sqrt{\frac{2}{s}}}{\sqrt{\frac{s}{2}}\left(1+\sqrt{\frac{s}{2}}\tan^{-1}\sqrt{\frac{2}{s}}\right)}$$
Inverting these LT's seems to be at least upon a first glance, nontrivial, however a numerical inversion is always possible.