An agrarian economy will naturally have a higher birth rate in part because children have a greater economic value and utility in an agrarian economy vs a post-industrial economy, where children can quickly become an economic burden rather than an asset.
Third World cultures are likely to be less self-oriented and individualistic while being more religious and traditional and therefore have a higher birth rate. First World countries center around the individual and his or her sense of satisfaction. Consumerism has made satisfying individual needs into a ritual and the individual into the totem idol in the center of his own worship circle. This leaves little room for children and abortion is as much the bastard child of consumerism, as fast food and 300 television channels.
Child labor laws have drastically decreased the economic value of children, (except as child actors and celebrities), accordingly birth rates have dropped even in industrialized Second World nations with child labor laws. With children less able to contribute to the family and with an increasingly prolonged educational process that now culminates anywhere from ages 18 to 22 to a high of 26, high birth rates become increasingly difficult to support. Consumerism itself drastically raises spending on children and by children, making larger families all the more economically trying.
When Third World cultures enter First World nations, they enter an industrial economy whose birth rate they quickly exceed. These multiple births however are out of place in an industrial economy which is no longer meant to accommodate them. Supporting such a birth rate therefore becomes a problem that is foisted onto social services. Once a family becomes dependent on social welfare programs, it can often decide to milk the system for all it can get, creating endemic fraud. Even if does not go this route however, the children are often put to finding extralegal means of contributing to the family income, which generates crime and creates a criminal culture within the family and the community that can then last for generations.
There are a number of ways for First World nations to cope with this situation.
First of all set immigration quotas by birth rate. While some immigrants are needed, immigrants who outreproduce you, are not, since they create radical change that a nation's own culture cannot sustain. The melting pot only melts so far until it explodes.
If immigration quotas are set by family size and by the average birth rate of the country of origin, a country can better accommodate the collision of birth rates, taking in those immigrants whose birth rate is more compatible with their adopted country.
This would for example mean more Polish (10/1000), Chinese (13/1000) and Irish (14/1000) immigrants and less Pakistani (27/1000), Mexican (20/1000) and Somalian (45/1000) ones. Note how this compares to the American (14/1000), Canadian (11/1000), British (10/1000), French (13/1000) birth rates -- and decide for yourself which groups are more compatible birth rate wise.
The advantage of this is obvious. Large birth rates are more taxing and require heavy investment in schools and social services infrastructure. Which diminishes the value of the immigrants to their host society and creates a heavy burden.
Secondly, increase native birth rates to better match those of the incoming populations. The common approach has been to do this through tax incentives and similar programs, but that has been demonstrated to be unsatisfactory. First World countries have adopted economies that make childbearing a burden, even with incentives and generous maternity leave and similar programs, all that is accomplished is to make the burden a little more manageable.
It is important to recognize that the entire economic structure in the West, from consumerism to high tax rates to heavy debt to extended educational processes that primarily serve as an advanced educational form of babysitting, effectively make large families difficult to sustain. Restoring the birth rates we used to have will require deemphasizing consumerism and the debt economy, reducing the tax burden significantly on native born families and enabling the return of the extended family, trimming the extended educational process and accepting a larger role for children and teenagers in contributing to family finances. While none of this is likely to happen, it is worth noting in order to contemplate what it would take to compete in terms of birth rates with incoming immigrant populations.
Thirdly the economic structure of the population in the target country and the country of origin must be heavily weighed in considering prospective immigration. People from a country of origin who are not capable of properly functioning in a Western economy without becoming a burden on social services should not be admitted. Western countries have no shortage of shopkeepers that we constantly need to be importing more and more of them.
Taking in immigrants who cannot function economically and demographically in a way that contributes positively to the country without excessive negatives, is unhelpful both to the host country and to the immigrants themselves.
The real problems of immigration stem from its unplanned and chaotic nature as random rules and a complete lack of oversight make no coherent attempt to plan the flow of immigration with a view to integration.