Acquisitive growth is the opposite of organic growth. It is the element of growth (including declines) that comes from buying and selling businesses.
It is important to separate organic and acquisition driven growth, as organic growth is more valuable. Any company can grow by buying other businesses as long as it has the funds (the fact that it can pay with shares makes this probable).
Organic growth is harder to achieve but usually more sustainable. Acquisitive growth also tends to be more expensive: acquisitions have to be paid for and the previous owners will usually manage to get a fair price or better. This means that acquisitions are rarely of much benefit to the shareholders of the acquiring company.
Several studies have shown that most takeovers destroy shareholder value.
There are some cases where acquisitive growth is good. In some industries companies have grown over the long term by making regular (usually small) acquisitions with strong synergies.
A good example of sustainable acquisitive growth is the pub sub-sector in the UK from the early 1990s onwards. There were large numbers of small chains and standalone pubs available. By buying these large chains could increase their purchasing power (cutting the cost of supplies) and centralise administration (cutting overheads).